GAP (MonoidalCategories) - Chapter 1: Monoidal Categories

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1 Monoidal Categories
1.1 Monoidal Categories
1.1-1 TensorProductOnMorphisms
1.1-2 TensorProductOnMorphismsWithGivenTensorProducts
1.1-3 AssociatorRightToLeft
1.1-4 AssociatorRightToLeftWithGivenTensorProducts
1.1-5 AssociatorLeftToRight
1.1-6 AssociatorLeftToRightWithGivenTensorProducts
1.1-7 LeftUnitor
1.1-8 LeftUnitorWithGivenTensorProduct
1.1-9 LeftUnitorInverse
1.1-10 LeftUnitorInverseWithGivenTensorProduct
1.1-11 RightUnitor
1.1-12 RightUnitorWithGivenTensorProduct
1.1-13 RightUnitorInverse
1.1-14 RightUnitorInverseWithGivenTensorProduct
1.1-15 TensorProductOnObjects
1.1-16 AddTensorProductOnObjects
1.1-17 TensorUnit
1.1-18 AddTensorUnit
1.2 Additive Monoidal Categories
1.2-1 LeftDistributivityExpanding
1.2-2 LeftDistributivityExpandingWithGivenObjects
1.2-3 LeftDistributivityFactoring
1.2-4 LeftDistributivityFactoringWithGivenObjects
1.2-5 RightDistributivityExpanding
1.2-6 RightDistributivityExpandingWithGivenObjects
1.2-7 RightDistributivityFactoring
1.2-8 RightDistributivityFactoringWithGivenObjects
1.3 Braided Monoidal Categories
1.3-1 Braiding
1.3-2 BraidingWithGivenTensorProducts
1.3-3 BraidingInverse
1.3-4 BraidingInverseWithGivenTensorProducts
1.4 Symmetric Monoidal Categories
1.5 Closed Monoidal Categories
1.5-1 InternalHomOnObjects
1.5-2 InternalHomOnMorphisms
1.5-3 InternalHomOnMorphismsWithGivenInternalHoms
1.5-4 EvaluationMorphism
1.5-5 EvaluationMorphismWithGivenSource
1.5-6 CoevaluationMorphism
1.5-7 CoevaluationMorphismWithGivenRange
1.5-8 TensorProductToInternalHomAdjunctionMap
1.5-9 TensorProductToInternalHomAdjunctionMapWithGivenInternalHom
1.5-10 InternalHomToTensorProductAdjunctionMap
1.5-11 InternalHomToTensorProductAdjunctionMapWithGivenTensorProduct
1.5-12 MonoidalPreComposeMorphism
1.5-13 MonoidalPreComposeMorphismWithGivenObjects
1.5-14 MonoidalPostComposeMorphism
1.5-15 MonoidalPostComposeMorphismWithGivenObjects
1.5-16 DualOnObjects
1.5-17 DualOnMorphisms
1.5-18 DualOnMorphismsWithGivenDuals
1.5-19 EvaluationForDual
1.5-20 EvaluationForDualWithGivenTensorProduct
1.5-21 MorphismToBidual
1.5-22 MorphismToBidualWithGivenBidual
1.5-23 TensorProductInternalHomCompatibilityMorphism
1.5-24 TensorProductInternalHomCompatibilityMorphismWithGivenObjects
1.5-25 TensorProductDualityCompatibilityMorphism
1.5-26 TensorProductDualityCompatibilityMorphismWithGivenObjects
1.5-27 MorphismFromTensorProductToInternalHom
1.5-28 MorphismFromTensorProductToInternalHomWithGivenObjects
1.5-29 IsomorphismFromDualObjectToInternalHomIntoTensorUnit
1.5-30 IsomorphismFromInternalHomIntoTensorUnitToDualObject
1.5-31 UniversalPropertyOfDual
1.5-32 LambdaIntroduction
1.5-33 LambdaElimination
1.5-34 IsomorphismFromObjectToInternalHom
1.5-35 IsomorphismFromObjectToInternalHomWithGivenInternalHom
1.5-36 IsomorphismFromInternalHomToObject
1.5-37 IsomorphismFromInternalHomToObjectWithGivenInternalHom
1.6 Coclosed Monoidal Categories
1.6-1 InternalCoHomOnObjects
1.6-2 InternalCoHomOnMorphisms
1.6-3 InternalCoHomOnMorphismsWithGivenInternalCoHoms
1.6-4 CoclosedEvaluationMorphism
1.6-5 CoclosedEvaluationMorphismWithGivenRange
1.6-6 CoclosedCoevaluationMorphism
1.6-7 CoclosedCoevaluationMorphismWithGivenSource
1.6-8 TensorProductToInternalCoHomAdjunctionMap
1.6-9 TensorProductToInternalCoHomAdjunctionMapWithGivenInternalCoHom
1.6-10 InternalCoHomToTensorProductAdjunctionMap
1.6-11 InternalCoHomToTensorProductAdjunctionMapWithGivenTensorProduct
1.6-12 MonoidalPreCoComposeMorphism
1.6-13 MonoidalPreCoComposeMorphismWithGivenObjects
1.6-14 MonoidalPostCoComposeMorphism
1.6-15 MonoidalPostCoComposeMorphismWithGivenObjects
1.6-16 CoDualOnObjects
1.6-17 CoDualOnMorphisms
1.6-18 CoDualOnMorphismsWithGivenCoDuals
1.6-19 CoclosedEvaluationForCoDual
1.6-20 CoclosedEvaluationForCoDualWithGivenTensorProduct
1.6-21 MorphismFromCoBidual
1.6-22 MorphismFromCoBidualWithGivenCoBidual
1.6-23 InternalCoHomTensorProductCompatibilityMorphism
1.6-24 InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects
1.6-25 CoDualityTensorProductCompatibilityMorphism
1.6-26 CoDualityTensorProductCompatibilityMorphismWithGivenObjects
1.6-27 MorphismFromInternalCoHomToTensorProduct
1.6-28 MorphismFromInternalCoHomToTensorProductWithGivenObjects
1.6-29 IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit
1.6-30 IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject
1.6-31 UniversalPropertyOfCoDual
1.6-32 CoLambdaIntroduction
1.6-33 CoLambdaElimination
1.6-34 IsomorphismFromObjectToInternalCoHom
1.6-35 IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom
1.6-36 IsomorphismFromInternalCoHomToObject
1.6-37 IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom
1.7 Symmetric Closed Monoidal Categories
1.8 Symmetric Coclosed Monoidal Categories
1.9 Rigid Symmetric Closed Monoidal Categories
1.9-1 IsomorphismFromTensorProductWithDualObjectToInternalHom
1.9-2 IsomorphismFromInternalHomToTensorProductWithDualObject
1.9-3 MorphismFromInternalHomToTensorProduct
1.9-4 MorphismFromInternalHomToTensorProductWithGivenObjects
1.9-5 TensorProductInternalHomCompatibilityMorphismInverse
1.9-6 TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects
1.9-7 CoevaluationForDual
1.9-8 CoevaluationForDualWithGivenTensorProduct
1.9-9 TraceMap
1.9-10 RankMorphism
1.9-11 MorphismFromBidual
1.9-12 MorphismFromBidualWithGivenBidual
1.10 Rigid Symmetric Coclosed Monoidal Categories
1.10-1 IsomorphismFromInternalCoHomToTensorProductWithCoDualObject
1.10-2 IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom
1.10-3 MorphismFromTensorProductToInternalCoHom
1.10-4 MorphismFromTensorProductToInternalCoHomWithGivenObjects
1.10-5 InternalCoHomTensorProductCompatibilityMorphismInverse
1.10-6 InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects
1.10-7 CoclosedCoevaluationForCoDual
1.10-8 CoclosedCoevaluationForCoDualWithGivenTensorProduct
1.10-9 CoTraceMap
1.10-10 CoRankMorphism
1.10-11 MorphismToCoBidual
1.10-12 MorphismToCoBidualWithGivenCoBidual
1.11 Convenience Methods
1.11-1 InternalHom
1.11-2 InternalCoHom
1.12 Add-methods
1.12-1 AddLeftDistributivityExpanding
1.12-2 AddLeftDistributivityExpandingWithGivenObjects
1.12-3 AddLeftDistributivityFactoring
1.12-4 AddLeftDistributivityFactoringWithGivenObjects
1.12-5 AddRightDistributivityExpanding
1.12-6 AddRightDistributivityExpandingWithGivenObjects
1.12-7 AddRightDistributivityFactoring
1.12-8 AddRightDistributivityFactoringWithGivenObjects
1.12-9 AddBraiding
1.12-10 AddBraidingInverse
1.12-11 AddBraidingInverseWithGivenTensorProducts
1.12-12 AddBraidingWithGivenTensorProducts
1.12-13 AddCoevaluationMorphism
1.12-14 AddCoevaluationMorphismWithGivenRange
1.12-15 AddDualOnMorphisms
1.12-16 AddDualOnMorphismsWithGivenDuals
1.12-17 AddDualOnObjects
1.12-18 AddEvaluationForDual
1.12-19 AddEvaluationForDualWithGivenTensorProduct
1.12-20 AddEvaluationMorphism
1.12-21 AddEvaluationMorphismWithGivenSource
1.12-22 AddInternalHomOnMorphisms
1.12-23 AddInternalHomOnMorphismsWithGivenInternalHoms
1.12-24 AddInternalHomOnObjects
1.12-25 AddInternalHomToTensorProductAdjunctionMap
1.12-26 AddInternalHomToTensorProductAdjunctionMapWithGivenTensorProduct
1.12-27 AddIsomorphismFromDualObjectToInternalHomIntoTensorUnit
1.12-28 AddIsomorphismFromInternalHomIntoTensorUnitToDualObject
1.12-29 AddIsomorphismFromInternalHomToObject
1.12-30 AddIsomorphismFromInternalHomToObjectWithGivenInternalHom
1.12-31 AddIsomorphismFromObjectToInternalHom
1.12-32 AddIsomorphismFromObjectToInternalHomWithGivenInternalHom
1.12-33 AddLambdaElimination
1.12-34 AddLambdaIntroduction
1.12-35 AddMonoidalPostComposeMorphism
1.12-36 AddMonoidalPostComposeMorphismWithGivenObjects
1.12-37 AddMonoidalPreComposeMorphism
1.12-38 AddMonoidalPreComposeMorphismWithGivenObjects
1.12-39 AddMorphismFromTensorProductToInternalHom
1.12-40 AddMorphismFromTensorProductToInternalHomWithGivenObjects
1.12-41 AddMorphismToBidual
1.12-42 AddMorphismToBidualWithGivenBidual
1.12-43 AddTensorProductDualityCompatibilityMorphism
1.12-44 AddTensorProductDualityCompatibilityMorphismWithGivenObjects
1.12-45 AddTensorProductInternalHomCompatibilityMorphism
1.12-46 AddTensorProductInternalHomCompatibilityMorphismWithGivenObjects
1.12-47 AddTensorProductToInternalHomAdjunctionMap
1.12-48 AddTensorProductToInternalHomAdjunctionMapWithGivenInternalHom
1.12-49 AddUniversalPropertyOfDual
1.12-50 AddCoDualOnMorphisms
1.12-51 AddCoDualOnMorphismsWithGivenCoDuals
1.12-52 AddCoDualOnObjects
1.12-53 AddCoDualityTensorProductCompatibilityMorphism
1.12-54 AddCoDualityTensorProductCompatibilityMorphismWithGivenObjects
1.12-55 AddCoLambdaElimination
1.12-56 AddCoLambdaIntroduction
1.12-57 AddCoclosedCoevaluationMorphism
1.12-58 AddCoclosedCoevaluationMorphismWithGivenSource
1.12-59 AddCoclosedEvaluationForCoDual
1.12-60 AddCoclosedEvaluationForCoDualWithGivenTensorProduct
1.12-61 AddCoclosedEvaluationMorphism
1.12-62 AddCoclosedEvaluationMorphismWithGivenRange
1.12-63 AddInternalCoHomOnMorphisms
1.12-64 AddInternalCoHomOnMorphismsWithGivenInternalCoHoms
1.12-65 AddInternalCoHomOnObjects
1.12-66 AddInternalCoHomTensorProductCompatibilityMorphism
1.12-67 AddInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects
1.12-68 AddInternalCoHomToTensorProductAdjunctionMap
1.12-69 AddInternalCoHomToTensorProductAdjunctionMapWithGivenTensorProduct
1.12-70 AddIsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit
1.12-71 AddIsomorphismFromInternalCoHomFromTensorUnitToCoDualObject
1.12-72 AddIsomorphismFromInternalCoHomToObject
1.12-73 AddIsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom
1.12-74 AddIsomorphismFromObjectToInternalCoHom
1.12-75 AddIsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom
1.12-76 AddMonoidalPostCoComposeMorphism
1.12-77 AddMonoidalPostCoComposeMorphismWithGivenObjects
1.12-78 AddMonoidalPreCoComposeMorphism
1.12-79 AddMonoidalPreCoComposeMorphismWithGivenObjects
1.12-80 AddMorphismFromCoBidual
1.12-81 AddMorphismFromCoBidualWithGivenCoBidual
1.12-82 AddMorphismFromInternalCoHomToTensorProduct
1.12-83 AddMorphismFromInternalCoHomToTensorProductWithGivenObjects
1.12-84 AddTensorProductToInternalCoHomAdjunctionMap
1.12-85 AddTensorProductToInternalCoHomAdjunctionMapWithGivenInternalCoHom
1.12-86 AddUniversalPropertyOfCoDual
1.12-87 AddAssociatorLeftToRight
1.12-88 AddAssociatorLeftToRightWithGivenTensorProducts
1.12-89 AddAssociatorRightToLeft
1.12-90 AddAssociatorRightToLeftWithGivenTensorProducts
1.12-91 AddLeftUnitor
1.12-92 AddLeftUnitorInverse
1.12-93 AddLeftUnitorInverseWithGivenTensorProduct
1.12-94 AddLeftUnitorWithGivenTensorProduct
1.12-95 AddRightUnitor
1.12-96 AddRightUnitorInverse
1.12-97 AddRightUnitorInverseWithGivenTensorProduct
1.12-98 AddRightUnitorWithGivenTensorProduct
1.12-99 AddTensorProductOnMorphisms
1.12-100 AddTensorProductOnMorphismsWithGivenTensorProducts
1.12-101 AddCoevaluationForDual
1.12-102 AddCoevaluationForDualWithGivenTensorProduct
1.12-103 AddIsomorphismFromInternalHomToTensorProductWithDualObject
1.12-104 AddIsomorphismFromTensorProductWithDualObjectToInternalHom
1.12-105 AddMorphismFromBidual
1.12-106 AddMorphismFromBidualWithGivenBidual
1.12-107 AddMorphismFromInternalHomToTensorProduct
1.12-108 AddMorphismFromInternalHomToTensorProductWithGivenObjects
1.12-109 AddRankMorphism
1.12-110 AddTensorProductInternalHomCompatibilityMorphismInverse
1.12-111 AddTensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects
1.12-112 AddTraceMap
1.12-113 AddCoRankMorphism
1.12-114 AddCoTraceMap
1.12-115 AddCoclosedCoevaluationForCoDual
1.12-116 AddCoclosedCoevaluationForCoDualWithGivenTensorProduct
1.12-117 AddInternalCoHomTensorProductCompatibilityMorphismInverse
1.12-118 AddInternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects
1.12-119 AddIsomorphismFromInternalCoHomToTensorProductWithCoDualObject
1.12-120 AddIsomorphismFromTensorProductWithCoDualObjectToInternalCoHom
1.12-121 AddMorphismFromTensorProductToInternalCoHom
1.12-122 AddMorphismFromTensorProductToInternalCoHomWithGivenObjects
1.12-123 AddMorphismToCoBidual
1.12-124 AddMorphismToCoBidualWithGivenCoBidual

### 1 Monoidal Categories

#### 1.1 Monoidal Categories

A 6-tuple ( \mathbf{C}, \otimes, 1, \alpha, \lambda, \rho ) consisting of

- a category \mathbf{C},

- a functor \otimes: \mathbf{C} \times \mathbf{C} \rightarrow \mathbf{C} compatible with the congruence of morphisms,

- an object 1 \in \mathbf{C},

- a natural isomorphism \alpha_{a,b,c}: a \otimes (b \otimes c) \cong (a \otimes b) \otimes c,

- a natural isomorphism \lambda_{a}: 1 \otimes a \cong a,

- a natural isomorphism \rho_{a}: a \otimes 1 \cong a,

is called a monoidal category, if

- for all objects a,b,c,d, the pentagon identity holds:

(\alpha_{a,b,c} \otimes \mathrm{id}_d) \circ \alpha_{a,b \otimes c, d} \circ ( \mathrm{id}_a \otimes \alpha_{b,c,d} ) \sim \alpha_{a \otimes b, c, d} \circ \alpha_{a,b,c \otimes d},

- for all objects a,c, the triangle identity holds:

( \rho_a \otimes \mathrm{id}_c ) \circ \alpha_{a,1,c} \sim \mathrm{id}_a \otimes \lambda_c.

The corresponding GAP property is given by `IsMonoidalCategory`.

##### 1.1-1 TensorProductOnMorphisms

`‣ TensorProductOnMorphisms`( alpha, beta ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a \otimes b, a' \otimes b')

The arguments are two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b'. The output is the tensor product \alpha \otimes \beta.

##### 1.1-2 TensorProductOnMorphismsWithGivenTensorProducts

`‣ TensorProductOnMorphismsWithGivenTensorProducts`( s, alpha, beta, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a \otimes b, a' \otimes b')

The arguments are an object s = a \otimes b, two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b', and an object r = a' \otimes b'. The output is the tensor product \alpha \otimes \beta.

##### 1.1-3 AssociatorRightToLeft

`‣ AssociatorRightToLeft`( a, b, c ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a \otimes (b \otimes c), (a \otimes b) \otimes c ).

The arguments are three objects a,b,c. The output is the associator \alpha_{a,(b,c)}: a \otimes (b \otimes c) \rightarrow (a \otimes b) \otimes c.

##### 1.1-4 AssociatorRightToLeftWithGivenTensorProducts

`‣ AssociatorRightToLeftWithGivenTensorProducts`( s, a, b, c, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a \otimes (b \otimes c), (a \otimes b) \otimes c ).

The arguments are an object s = a \otimes (b \otimes c), three objects a,b,c, and an object r = (a \otimes b) \otimes c. The output is the associator \alpha_{a,(b,c)}: a \otimes (b \otimes c) \rightarrow (a \otimes b) \otimes c.

##### 1.1-5 AssociatorLeftToRight

`‣ AssociatorLeftToRight`( a, b, c ) | ( operation )

Returns: a morphism in \mathrm{Hom}( (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c) ).

The arguments are three objects a,b,c. The output is the associator \alpha_{(a,b),c}: (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c).

##### 1.1-6 AssociatorLeftToRightWithGivenTensorProducts

`‣ AssociatorLeftToRightWithGivenTensorProducts`( s, a, b, c, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c) ).

The arguments are an object s = (a \otimes b) \otimes c, three objects a,b,c, and an object r = a \otimes (b \otimes c). The output is the associator \alpha_{(a,b),c}: (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c).

##### 1.1-7 LeftUnitor

`‣ LeftUnitor`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(1 \otimes a, a)

The argument is an object a. The output is the left unitor \lambda_a: 1 \otimes a \rightarrow a.

##### 1.1-8 LeftUnitorWithGivenTensorProduct

`‣ LeftUnitorWithGivenTensorProduct`( a, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}(1 \otimes a, a)

The arguments are an object a and an object s = 1 \otimes a. The output is the left unitor \lambda_a: 1 \otimes a \rightarrow a.

##### 1.1-9 LeftUnitorInverse

`‣ LeftUnitorInverse`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a, 1 \otimes a)

The argument is an object a. The output is the inverse of the left unitor \lambda_a^{-1}: a \rightarrow 1 \otimes a.

##### 1.1-10 LeftUnitorInverseWithGivenTensorProduct

`‣ LeftUnitorInverseWithGivenTensorProduct`( a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a, 1 \otimes a)

The argument is an object a and an object r = 1 \otimes a. The output is the inverse of the left unitor \lambda_a^{-1}: a \rightarrow 1 \otimes a.

##### 1.1-11 RightUnitor

`‣ RightUnitor`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a \otimes 1, a)

The argument is an object a. The output is the right unitor \rho_a: a \otimes 1 \rightarrow a.

##### 1.1-12 RightUnitorWithGivenTensorProduct

`‣ RightUnitorWithGivenTensorProduct`( a, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a \otimes 1, a)

The arguments are an object a and an object s = a \otimes 1. The output is the right unitor \rho_a: a \otimes 1 \rightarrow a.

##### 1.1-13 RightUnitorInverse

`‣ RightUnitorInverse`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a, a \otimes 1)

The argument is an object a. The output is the inverse of the right unitor \rho_a^{-1}: a \rightarrow a \otimes 1.

##### 1.1-14 RightUnitorInverseWithGivenTensorProduct

`‣ RightUnitorInverseWithGivenTensorProduct`( a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a, a \otimes 1)

The arguments are an object a and an object r = a \otimes 1. The output is the inverse of the right unitor \rho_a^{-1}: a \rightarrow a \otimes 1.

##### 1.1-15 TensorProductOnObjects

`‣ TensorProductOnObjects`( a, b ) | ( operation )

Returns: an object

The arguments are two objects a, b. The output is the tensor product a \otimes b.

##### 1.1-16 AddTensorProductOnObjects

`‣ AddTensorProductOnObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductOnObjects`. F: (a,b) \mapsto a \otimes b.

##### 1.1-17 TensorUnit

`‣ TensorUnit`( C ) | ( attribute )

Returns: an object

The argument is a category \mathbf{C}. The output is the tensor unit 1 of \mathbf{C}.

##### 1.1-18 AddTensorUnit

`‣ AddTensorUnit`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorUnit`. F: ( ) \mapsto 1.

#### 1.2 Additive Monoidal Categories

##### 1.2-1 LeftDistributivityExpanding

`‣ LeftDistributivityExpanding`( a, L ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a \otimes (b_1 \oplus \dots \oplus b_n), (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n) )

The arguments are an object a and a list of objects L = (b_1, \dots, b_n). The output is the left distributivity morphism a \otimes (b_1 \oplus \dots \oplus b_n) \rightarrow (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n).

##### 1.2-2 LeftDistributivityExpandingWithGivenObjects

`‣ LeftDistributivityExpandingWithGivenObjects`( s, a, L, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r )

The arguments are an object s = a \otimes (b_1 \oplus \dots \oplus b_n), an object a, a list of objects L = (b_1, \dots, b_n), and an object r = (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n). The output is the left distributivity morphism s \rightarrow r.

##### 1.2-3 LeftDistributivityFactoring

`‣ LeftDistributivityFactoring`( a, L ) | ( operation )

Returns: a morphism in \mathrm{Hom}( (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n), a \otimes (b_1 \oplus \dots \oplus b_n) )

The arguments are an object a and a list of objects L = (b_1, \dots, b_n). The output is the left distributivity morphism (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n) \rightarrow a \otimes (b_1 \oplus \dots \oplus b_n).

##### 1.2-4 LeftDistributivityFactoringWithGivenObjects

`‣ LeftDistributivityFactoringWithGivenObjects`( s, a, L, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r )

The arguments are an object s = (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n), an object a, a list of objects L = (b_1, \dots, b_n), and an object r = a \otimes (b_1 \oplus \dots \oplus b_n). The output is the left distributivity morphism s \rightarrow r.

##### 1.2-5 RightDistributivityExpanding

`‣ RightDistributivityExpanding`( L, a ) | ( operation )

Returns: a morphism in \mathrm{Hom}( (b_1 \oplus \dots \oplus b_n) \otimes a, (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a) )

The arguments are a list of objects L = (b_1, \dots, b_n) and an object a. The output is the right distributivity morphism (b_1 \oplus \dots \oplus b_n) \otimes a \rightarrow (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a).

##### 1.2-6 RightDistributivityExpandingWithGivenObjects

`‣ RightDistributivityExpandingWithGivenObjects`( s, L, a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r )

The arguments are an object s = (b_1 \oplus \dots \oplus b_n) \otimes a, a list of objects L = (b_1, \dots, b_n), an object a, and an object r = (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a). The output is the right distributivity morphism s \rightarrow r.

##### 1.2-7 RightDistributivityFactoring

`‣ RightDistributivityFactoring`( L, a ) | ( operation )

Returns: a morphism in \mathrm{Hom}( (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a), (b_1 \oplus \dots \oplus b_n) \otimes a)

The arguments are a list of objects L = (b_1, \dots, b_n) and an object a. The output is the right distributivity morphism (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a) \rightarrow (b_1 \oplus \dots \oplus b_n) \otimes a .

##### 1.2-8 RightDistributivityFactoringWithGivenObjects

`‣ RightDistributivityFactoringWithGivenObjects`( s, L, a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( s, r )

The arguments are an object s = (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a), a list of objects L = (b_1, \dots, b_n), an object a, and an object r = (b_1 \oplus \dots \oplus b_n) \otimes a. The output is the right distributivity morphism s \rightarrow r.

#### 1.3 Braided Monoidal Categories

A monoidal category \mathbf{C} equipped with a natural isomorphism B_{a,b}: a \otimes b \cong b \otimes a is called a braided monoidal category if

- \lambda_a \circ B_{a,1} \sim \rho_a,

- (B_{c,a} \otimes \mathrm{id}_b) \circ \alpha_{c,a,b} \circ B_{a \otimes b,c} \sim \alpha_{a,c,b} \circ ( \mathrm{id}_a \otimes B_{b,c}) \circ \alpha^{-1}_{a,b,c},

- ( \mathrm{id}_b \otimes B_{c,a} ) \circ \alpha^{-1}_{b,c,a} \circ B_{a,b \otimes c} \sim \alpha^{-1}_{b,a,c} \circ (B_{a,b} \otimes \mathrm{id}_c) \circ \alpha_{a,b,c}.

The corresponding GAP property is given by `IsBraidedMonoidalCategory`.

##### 1.3-1 Braiding

`‣ Braiding`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a \otimes b, b \otimes a ).

The arguments are two objects a,b. The output is the braiding B_{a,b}: a \otimes b \rightarrow b \otimes a.

##### 1.3-2 BraidingWithGivenTensorProducts

`‣ BraidingWithGivenTensorProducts`( s, a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a \otimes b, b \otimes a ).

The arguments are an object s = a \otimes b, two objects a,b, and an object r = b \otimes a. The output is the braiding B_{a,b}: a \otimes b \rightarrow b \otimes a.

##### 1.3-3 BraidingInverse

`‣ BraidingInverse`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( b \otimes a, a \otimes b ).

The arguments are two objects a,b. The output is the inverse braiding B_{a,b}^{-1}: b \otimes a \rightarrow a \otimes b.

##### 1.3-4 BraidingInverseWithGivenTensorProducts

`‣ BraidingInverseWithGivenTensorProducts`( s, a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( b \otimes a, a \otimes b ).

The arguments are an object s = b \otimes a, two objects a,b, and an object r = a \otimes b. The output is the inverse braiding B_{a,b}^{-1}: b \otimes a \rightarrow a \otimes b.

#### 1.4 Symmetric Monoidal Categories

A braided monoidal category \mathbf{C} is called symmetric monoidal category if B_{a,b}^{-1} \sim B_{b,a}. The corresponding GAP property is given by `IsSymmetricMonoidalCategory`.

#### 1.5 Closed Monoidal Categories

A monoidal category \mathbf{C} which has for each functor - \otimes b: \mathbf{C} \rightarrow \mathbf{C} a right adjoint (denoted by \mathrm{\underline{Hom}}(b,-)) is called a closed monoidal category.

If no operations involving duals are installed manually, the dual objects will be derived as a^\vee \coloneqq \mathrm{\underline{Hom}}(a,1).

The corresponding GAP property is called `IsClosedMonoidalCategory`.

##### 1.5-1 InternalHomOnObjects

`‣ InternalHomOnObjects`( a, b ) | ( operation )

Returns: an object

The arguments are two objects a,b. The output is the internal hom object \mathrm{\underline{Hom}}(a,b).

##### 1.5-2 InternalHomOnMorphisms

`‣ InternalHomOnMorphisms`( alpha, beta ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(a',b), \mathrm{\underline{Hom}}(a,b') )

The arguments are two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b'. The output is the internal hom morphism \mathrm{\underline{Hom}}(\alpha,\beta): \mathrm{\underline{Hom}}(a',b) \rightarrow \mathrm{\underline{Hom}}(a,b').

##### 1.5-3 InternalHomOnMorphismsWithGivenInternalHoms

`‣ InternalHomOnMorphismsWithGivenInternalHoms`( s, alpha, beta, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(a',b), \mathrm{\underline{Hom}}(a,b') )

The arguments are an object s = \mathrm{\underline{Hom}}(a',b), two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b', and an object r = \mathrm{\underline{Hom}}(a,b'). The output is the internal hom morphism \mathrm{\underline{Hom}}(\alpha,\beta): \mathrm{\underline{Hom}}(a',b) \rightarrow \mathrm{\underline{Hom}}(a,b').

##### 1.5-4 EvaluationMorphism

`‣ EvaluationMorphism`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(a,b) \otimes a, b ).

The arguments are two objects a, b. The output is the evaluation morphism \mathrm{ev}_{a,b}: \mathrm{\underline{Hom}}(a,b) \otimes a \rightarrow b, i.e., the counit of the tensor hom adjunction.

##### 1.5-5 EvaluationMorphismWithGivenSource

`‣ EvaluationMorphismWithGivenSource`( a, b, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(a,b) \otimes a, b ).

The arguments are two objects a,b and an object s = \mathrm{\underline{Hom}}(a,b) \otimes a. The output is the evaluation morphism \mathrm{ev}_{a,b}: \mathrm{\underline{Hom}}(a,b) \otimes a \rightarrow b, i.e., the counit of the tensor hom adjunction.

##### 1.5-6 CoevaluationMorphism

`‣ CoevaluationMorphism`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a, \mathrm{\underline{Hom}}(b, a \otimes b) ).

The arguments are two objects a,b. The output is the coevaluation morphism \mathrm{coev}_{a,b}: a \rightarrow \mathrm{\underline{Hom}}(b, a \otimes b), i.e., the unit of the tensor hom adjunction.

##### 1.5-7 CoevaluationMorphismWithGivenRange

`‣ CoevaluationMorphismWithGivenRange`( a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a, \mathrm{\underline{Hom}}(b, a \otimes b) ).

The arguments are two objects a,b and an object r = \mathrm{\underline{Hom}}(b, a \otimes b). The output is the coevaluation morphism \mathrm{coev}_{a,b}: a \rightarrow \mathrm{\underline{Hom}}(b, a \otimes b), i.e., the unit of the tensor hom adjunction.

##### 1.5-8 TensorProductToInternalHomAdjunctionMap

`‣ TensorProductToInternalHomAdjunctionMap`( a, b, f ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a, \mathrm{\underline{Hom}}(b,c) ).

The arguments are two objects a,b and a morphism f: a \otimes b \rightarrow c. The output is a morphism g: a \rightarrow \mathrm{\underline{Hom}}(b,c) corresponding to f under the tensor hom adjunction.

##### 1.5-9 TensorProductToInternalHomAdjunctionMapWithGivenInternalHom

`‣ TensorProductToInternalHomAdjunctionMapWithGivenInternalHom`( a, b, f, i ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a, \mathrm{\underline{Hom}}(b,c) ).

The arguments are two objects a,b, a morphism f: a \otimes b \rightarrow c and an object i = \mathrm{\underline{Hom}}(b,c). The output is a morphism g: a \rightarrow \mathrm{\underline{Hom}}(b,c) corresponding to f under the tensor hom adjunction.

##### 1.5-10 InternalHomToTensorProductAdjunctionMap

`‣ InternalHomToTensorProductAdjunctionMap`( b, c, g ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a \otimes b, c).

The arguments are two objects b,c and a morphism g: a \rightarrow \mathrm{\underline{Hom}}(b,c). The output is a morphism f: a \otimes b \rightarrow c corresponding to g under the tensor hom adjunction.

##### 1.5-11 InternalHomToTensorProductAdjunctionMapWithGivenTensorProduct

`‣ InternalHomToTensorProductAdjunctionMapWithGivenTensorProduct`( b, c, g, t ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a \otimes b, c).

The arguments are two objects b,c, a morphism g: a \rightarrow \mathrm{\underline{Hom}}(b,c) and an object t = a \otimes b. The output is a morphism f: a \otimes b \rightarrow c corresponding to g under the tensor hom adjunction.

##### 1.5-12 MonoidalPreComposeMorphism

`‣ MonoidalPreComposeMorphism`( a, b, c ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c), \mathrm{\underline{Hom}}(a,c) ).

The arguments are three objects a,b,c. The output is the precomposition morphism \mathrm{MonoidalPreComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c) \rightarrow \mathrm{\underline{Hom}}(a,c).

##### 1.5-13 MonoidalPreComposeMorphismWithGivenObjects

`‣ MonoidalPreComposeMorphismWithGivenObjects`( s, a, b, c, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c), \mathrm{\underline{Hom}}(a,c) ).

The arguments are an object s = \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c), three objects a,b,c, and an object r = \mathrm{\underline{Hom}}(a,c). The output is the precomposition morphism \mathrm{MonoidalPreComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c) \rightarrow \mathrm{\underline{Hom}}(a,c).

##### 1.5-14 MonoidalPostComposeMorphism

`‣ MonoidalPostComposeMorphism`( a, b, c ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b), \mathrm{\underline{Hom}}(a,c) ).

The arguments are three objects a,b,c. The output is the postcomposition morphism \mathrm{MonoidalPostComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b) \rightarrow \mathrm{\underline{Hom}}(a,c).

##### 1.5-15 MonoidalPostComposeMorphismWithGivenObjects

`‣ MonoidalPostComposeMorphismWithGivenObjects`( s, a, b, c, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b), \mathrm{\underline{Hom}}(a,c) ).

The arguments are an object s = \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b), three objects a,b,c, and an object r = \mathrm{\underline{Hom}}(a,c). The output is the postcomposition morphism \mathrm{MonoidalPostComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b) \rightarrow \mathrm{\underline{Hom}}(a,c).

##### 1.5-16 DualOnObjects

`‣ DualOnObjects`( a ) | ( attribute )

Returns: an object

The argument is an object a. The output is its dual object a^{\vee}.

##### 1.5-17 DualOnMorphisms

`‣ DualOnMorphisms`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}( b^{\vee}, a^{\vee} ).

The argument is a morphism \alpha: a \rightarrow b. The output is its dual morphism \alpha^{\vee}: b^{\vee} \rightarrow a^{\vee}.

##### 1.5-18 DualOnMorphismsWithGivenDuals

`‣ DualOnMorphismsWithGivenDuals`( s, alpha, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( b^{\vee}, a^{\vee} ).

The argument is an object s = b^{\vee}, a morphism \alpha: a \rightarrow b, and an object r = a^{\vee}. The output is the dual morphism \alpha^{\vee}: b^{\vee} \rightarrow a^{\vee}.

##### 1.5-19 EvaluationForDual

`‣ EvaluationForDual`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}( a^{\vee} \otimes a, 1 ).

The argument is an object a. The output is the evaluation morphism \mathrm{ev}_{a}: a^{\vee} \otimes a \rightarrow 1.

##### 1.5-20 EvaluationForDualWithGivenTensorProduct

`‣ EvaluationForDualWithGivenTensorProduct`( s, a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a^{\vee} \otimes a, 1 ).

The arguments are an object s = a^{\vee} \otimes a, an object a, and an object r = 1. The output is the evaluation morphism \mathrm{ev}_{a}: a^{\vee} \otimes a \rightarrow 1.

##### 1.5-21 MorphismToBidual

`‣ MorphismToBidual`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a, (a^{\vee})^{\vee}).

The argument is an object a. The output is the morphism to the bidual a \rightarrow (a^{\vee})^{\vee}.

##### 1.5-22 MorphismToBidualWithGivenBidual

`‣ MorphismToBidualWithGivenBidual`( a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a, (a^{\vee})^{\vee}).

The arguments are an object a, and an object r = (a^{\vee})^{\vee}. The output is the morphism to the bidual a \rightarrow (a^{\vee})^{\vee}.

##### 1.5-23 TensorProductInternalHomCompatibilityMorphism

`‣ TensorProductInternalHomCompatibilityMorphism`( list ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b'), \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b')).

The argument is a list of four objects [ a, a', b, b' ]. The output is the natural morphism \mathrm{TensorProductInternalHomCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b') \rightarrow \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b').

##### 1.5-24 TensorProductInternalHomCompatibilityMorphismWithGivenObjects

`‣ TensorProductInternalHomCompatibilityMorphismWithGivenObjects`( s, list, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b'), \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b')).

The arguments are a list of four objects [ a, a', b, b' ], and two objects s = \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b') and r = \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b'). The output is the natural morphism \mathrm{TensorProductInternalHomCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b') \rightarrow \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b').

##### 1.5-25 TensorProductDualityCompatibilityMorphism

`‣ TensorProductDualityCompatibilityMorphism`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a^{\vee} \otimes b^{\vee}, (a \otimes b)^{\vee} ).

The arguments are two objects a,b. The output is the natural morphism \mathrm{TensorProductDualityCompatibilityMorphismWithGivenObjects}: a^{\vee} \otimes b^{\vee} \rightarrow (a \otimes b)^{\vee}.

##### 1.5-26 TensorProductDualityCompatibilityMorphismWithGivenObjects

`‣ TensorProductDualityCompatibilityMorphismWithGivenObjects`( s, a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a^{\vee} \otimes b^{\vee}, (a \otimes b)^{\vee} ).

The arguments are an object s = a^{\vee} \otimes b^{\vee}, two objects a,b, and an object r = (a \otimes b)^{\vee}. The output is the natural morphism \mathrm{TensorProductDualityCompatibilityMorphismWithGivenObjects}_{a,b}: a^{\vee} \otimes b^{\vee} \rightarrow (a \otimes b)^{\vee}.

##### 1.5-27 MorphismFromTensorProductToInternalHom

`‣ MorphismFromTensorProductToInternalHom`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a^{\vee} \otimes b, \mathrm{\underline{Hom}}(a,b) ).

The arguments are two objects a,b. The output is the natural morphism \mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}}(a,b).

##### 1.5-28 MorphismFromTensorProductToInternalHomWithGivenObjects

`‣ MorphismFromTensorProductToInternalHomWithGivenObjects`( s, a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a^{\vee} \otimes b, \mathrm{\underline{Hom}}(a,b) ).

The arguments are an object s = a^{\vee} \otimes b, two objects a,b, and an object r = \mathrm{\underline{Hom}}(a,b). The output is the natural morphism \mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}}(a,b).

##### 1.5-29 IsomorphismFromDualObjectToInternalHomIntoTensorUnit

`‣ IsomorphismFromDualObjectToInternalHomIntoTensorUnit`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a^{\vee}, \mathrm{\underline{Hom}}(a,1)).

The argument is an object a. The output is the isomorphism \mathrm{IsomorphismFromDualObjectToInternalHomIntoTensorUnit}_{a}: a^{\vee} \rightarrow \mathrm{\underline{Hom}}(a,1).

##### 1.5-30 IsomorphismFromInternalHomIntoTensorUnitToDualObject

`‣ IsomorphismFromInternalHomIntoTensorUnitToDualObject`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(\mathrm{\underline{Hom}}(a,1), a^{\vee}).

The argument is an object a. The output is the isomorphism \mathrm{IsomorphismFromInternalHomIntoTensorUnitToDualObject}_{a}: \mathrm{\underline{Hom}}(a,1) \rightarrow a^{\vee}.

##### 1.5-31 UniversalPropertyOfDual

`‣ UniversalPropertyOfDual`( t, a, alpha ) | ( operation )

Returns: a morphism in \mathrm{Hom}(t, a^{\vee}).

The arguments are two objects t,a, and a morphism \alpha: t \otimes a \rightarrow 1. The output is the morphism t \rightarrow a^{\vee} given by the universal property of a^{\vee}.

##### 1.5-32 LambdaIntroduction

`‣ LambdaIntroduction`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}( 1, \mathrm{\underline{Hom}}(a,b) ).

The argument is a morphism \alpha: a \rightarrow b. The output is the corresponding morphism 1 \rightarrow \mathrm{\underline{Hom}}(a,b) under the tensor hom adjunction.

##### 1.5-33 LambdaElimination

`‣ LambdaElimination`( a, b, alpha ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a,b).

The arguments are two objects a,b, and a morphism \alpha: 1 \rightarrow \mathrm{\underline{Hom}}(a,b). The output is a morphism a \rightarrow b corresponding to \alpha under the tensor hom adjunction.

##### 1.5-34 IsomorphismFromObjectToInternalHom

`‣ IsomorphismFromObjectToInternalHom`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a, \mathrm{\underline{Hom}}(1,a)).

The argument is an object a. The output is the natural isomorphism a \rightarrow \mathrm{\underline{Hom}}(1,a).

##### 1.5-35 IsomorphismFromObjectToInternalHomWithGivenInternalHom

`‣ IsomorphismFromObjectToInternalHomWithGivenInternalHom`( a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a, \mathrm{\underline{Hom}}(1,a)).

The argument is an object a, and an object r = \mathrm{\underline{Hom}}(1,a). The output is the natural isomorphism a \rightarrow \mathrm{\underline{Hom}}(1,a).

##### 1.5-36 IsomorphismFromInternalHomToObject

`‣ IsomorphismFromInternalHomToObject`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(\mathrm{\underline{Hom}}(1,a),a).

The argument is an object a. The output is the natural isomorphism \mathrm{\underline{Hom}}(1,a) \rightarrow a.

##### 1.5-37 IsomorphismFromInternalHomToObjectWithGivenInternalHom

`‣ IsomorphismFromInternalHomToObjectWithGivenInternalHom`( a, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}(\mathrm{\underline{Hom}}(1,a),a).

The argument is an object a, and an object s = \mathrm{\underline{Hom}}(1,a). The output is the natural isomorphism \mathrm{\underline{Hom}}(1,a) \rightarrow a.

#### 1.6 Coclosed Monoidal Categories

A monoidal category \mathbf{C} which has for each functor - \otimes b: \mathbf{C} \rightarrow \mathbf{C} a left adjoint (denoted by \mathrm{\underline{coHom}}(-,b)) is called a coclosed monoidal category.

If no operations involving coduals are installed manually, the codual objects will be derived as a_\vee \coloneqq \mathrm{\underline{coHom}}(1,a).

The corresponding GAP property is called `IsCoclosedMonoidalCategory`.

##### 1.6-1 InternalCoHomOnObjects

`‣ InternalCoHomOnObjects`( a, b ) | ( operation )

Returns: an object

The arguments are two objects a,b. The output is the internal cohom object \mathrm{\underline{coHom}}(a,b).

##### 1.6-2 InternalCoHomOnMorphisms

`‣ InternalCoHomOnMorphisms`( alpha, beta ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b'), \mathrm{\underline{coHom}}(a',b) )

The arguments are two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b'. The output is the internal cohom morphism \mathrm{\underline{coHom}}(\alpha,\beta): \mathrm{\underline{coHom}}(a,b') \rightarrow \mathrm{\underline{coHom}}(a',b).

##### 1.6-3 InternalCoHomOnMorphismsWithGivenInternalCoHoms

`‣ InternalCoHomOnMorphismsWithGivenInternalCoHoms`( s, alpha, beta, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b'), \mathrm{\underline{coHom}}(a',b) )

The arguments are an object s = \mathrm{\underline{coHom}}(a,b'), two morphisms \alpha: a \rightarrow a', \beta: b \rightarrow b', and an object r = \mathrm{\underline{coHom}}(a',b). The output is the internal cohom morphism \mathrm{\underline{coHom}}(\alpha,\beta): \mathrm{\underline{coHom}}(a,b') \rightarrow \mathrm{\underline{coHom}}(a',b).

##### 1.6-4 CoclosedEvaluationMorphism

`‣ CoclosedEvaluationMorphism`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a, \mathrm{\underline{coHom}}(a,b) \otimes b ).

The arguments are two objects a, b. The output is the coclosed evaluation morphism \mathrm{coclev}_{a,b}: a \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes b, i.e., the unit of the cohom tensor adjunction.

##### 1.6-5 CoclosedEvaluationMorphismWithGivenRange

`‣ CoclosedEvaluationMorphismWithGivenRange`( a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a, \mathrm{\underline{coHom}}(a,b) \otimes b ).

The arguments are two objects a,b and an object r = \mathrm{\underline{coHom}}(a,b) \otimes b. The output is the coclosed evaluation morphism \mathrm{coclev}_{a,b}: a \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes b, i.e., the unit of the cohom tensor adjunction.

##### 1.6-6 CoclosedCoevaluationMorphism

`‣ CoclosedCoevaluationMorphism`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a \otimes b, b), a ).

The arguments are two objects a,b. The output is the coclosed coevaluation morphism \mathrm{coclcoev}_{a,b}: \mathrm{\underline{coHom}}(a \otimes b, b) \rightarrow a, i.e., the counit of the cohom tensor adjunction.

##### 1.6-7 CoclosedCoevaluationMorphismWithGivenSource

`‣ CoclosedCoevaluationMorphismWithGivenSource`( a, b, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a \otimes b, b), b ).

The arguments are two objects a,b and an object s = \mathrm{\underline{coHom}(a \otimes b, b)}. The output is the coclosed coevaluation morphism \mathrm{coclcoev}_{a,b}: \mathrm{\underline{coHom}}(a \otimes b, b) \rightarrow a, i.e., the unit of the cohom tensor adjunction.

##### 1.6-8 TensorProductToInternalCoHomAdjunctionMap

`‣ TensorProductToInternalCoHomAdjunctionMap`( c, b, g ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), c ).

The arguments are two objects c,b and a morphism g: a \rightarrow c \otimes b. The output is a morphism f: \mathrm{\underline{coHom}}(a,b) \rightarrow c corresponding to g under the cohom tensor adjunction.

##### 1.6-9 TensorProductToInternalCoHomAdjunctionMapWithGivenInternalCoHom

`‣ TensorProductToInternalCoHomAdjunctionMapWithGivenInternalCoHom`( c, b, g, i ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), c ).

The arguments are two objects c,b, a morphism g: a \rightarrow c \otimes b and an object i = \mathrm{\underline{coHom}(a,b)}. The output is a morphism f: \mathrm{\underline{coHom}}(a,b) \rightarrow c corresponding to g under the cohom tensor adjunction.

##### 1.6-10 InternalCoHomToTensorProductAdjunctionMap

`‣ InternalCoHomToTensorProductAdjunctionMap`( a, b, f ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a, c \otimes b).

The arguments are two objects a,b and a morphism f: \mathrm{\underline{coHom}}(a,b) \rightarrow c. The output is a morphism g: a \rightarrow c \otimes b corresponding to f under the cohom tensor adjunction.

##### 1.6-11 InternalCoHomToTensorProductAdjunctionMapWithGivenTensorProduct

`‣ InternalCoHomToTensorProductAdjunctionMapWithGivenTensorProduct`( a, b, f, t ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a, c \otimes b).

The arguments are two objects a,b, a morphism f: \mathrm{\underline{coHom}}(a,b) \rightarrow c and an object t = c \otimes b. The output is a morphism g: a \rightarrow c \otimes b corresponding to f under the cohom tensor adjunction.

##### 1.6-12 MonoidalPreCoComposeMorphism

`‣ MonoidalPreCoComposeMorphism`( a, b, c ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,c), \mathrm{\underline{coHom}}(b,c) \otimes \mathrm{\underline{coHom}}(a,b) ).

The arguments are three objects a,b,c. The output is the precocomposition morphism \mathrm{MonoidalPreCoComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{coHom}}(a,c) \rightarrow \mathrm{\underline{coHom}}(b,c) \otimes \mathrm{\underline{coHom}}(a,b).

##### 1.6-13 MonoidalPreCoComposeMorphismWithGivenObjects

`‣ MonoidalPreCoComposeMorphismWithGivenObjects`( s, a, b, c, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,c), \mathrm{\underline{coHom}}(b,c) \otimes \mathrm{\underline{coHom}}(a,b) ).

The arguments are an object s = \mathrm{\underline{coHom}}(a,c), three objects a,b,c, and an object r = \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(b,c). The output is the precocomposition morphism \mathrm{MonoidalPreCoComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{coHom}}(a,c) \rightarrow \mathrm{\underline{coHom}}(b,c) \otimes \mathrm{\underline{coHom}}(a,b).

##### 1.6-14 MonoidalPostCoComposeMorphism

`‣ MonoidalPostCoComposeMorphism`( a, b, c ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,c), \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(b,c) ).

The arguments are three objects a,b,c. The output is the postcocomposition morphism \mathrm{MonoidalPostCoComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{coHom}}(a,c) \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(b,c).

##### 1.6-15 MonoidalPostCoComposeMorphismWithGivenObjects

`‣ MonoidalPostCoComposeMorphismWithGivenObjects`( s, a, b, c, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,c), \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(b,c) ).

The arguments are an object s = \mathrm{\underline{coHom}}(a,c), three objects a,b,c, and an object r = \mathrm{\underline{coHom}}(b,c) \otimes \mathrm{\underline{coHom}}(a,b). The output is the postcocomposition morphism \mathrm{MonoidalPostCoComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{coHom}}(a,c) \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(b,c).

##### 1.6-16 CoDualOnObjects

`‣ CoDualOnObjects`( a ) | ( attribute )

Returns: an object

The argument is an object a. The output is its codual object a_{\vee}.

##### 1.6-17 CoDualOnMorphisms

`‣ CoDualOnMorphisms`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}( b_{\vee}, a_{\vee} ).

The argument is a morphism \alpha: a \rightarrow b. The output is its codual morphism \alpha_{\vee}: b_{\vee} \rightarrow a_{\vee}.

##### 1.6-18 CoDualOnMorphismsWithGivenCoDuals

`‣ CoDualOnMorphismsWithGivenCoDuals`( s, alpha, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( b_{\vee}, a_{\vee} ).

The argument is an object s = b_{\vee}, a morphism \alpha: a \rightarrow b, and an object r = a_{\vee}. The output is the dual morphism \alpha_{\vee}: b^{\vee} \rightarrow a^{\vee}.

##### 1.6-19 CoclosedEvaluationForCoDual

`‣ CoclosedEvaluationForCoDual`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}( 1, a_{\vee} \otimes a ).

The argument is an object a. The output is the coclosed evaluation morphism \mathrm{coclev}_{a}: 1 \rightarrow a_{\vee} \otimes a.

##### 1.6-20 CoclosedEvaluationForCoDualWithGivenTensorProduct

`‣ CoclosedEvaluationForCoDualWithGivenTensorProduct`( s, a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( 1, a_{\vee} \otimes a ).

The arguments are an object s = 1, an object a, and an object r = a_{\vee} \otimes a. The output is the coclosed evaluation morphism \mathrm{coclev}_{a}: 1 \rightarrow a_{\vee} \otimes a.

##### 1.6-21 MorphismFromCoBidual

`‣ MorphismFromCoBidual`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}((a_{\vee})_{\vee}, a).

The argument is an object a. The output is the morphism from the cobidual (a_{\vee})_{\vee} \rightarrow a.

##### 1.6-22 MorphismFromCoBidualWithGivenCoBidual

`‣ MorphismFromCoBidualWithGivenCoBidual`( a, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}((a_{\vee})_{\vee}, a).

The arguments are an object a, and an object s = (a_{\vee})_{\vee}. The output is the morphism from the cobidual (a_{\vee})_{\vee} \rightarrow a.

##### 1.6-23 InternalCoHomTensorProductCompatibilityMorphism

`‣ InternalCoHomTensorProductCompatibilityMorphism`( list ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a \otimes a', b \otimes b'), \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b')).

The argument is a list of four objects [ a, a', b, b' ]. The output is the natural morphism \mathrm{InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{coHom}}(a \otimes a', b \otimes b') \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b').

##### 1.6-24 InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects

`‣ InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects`( s, list, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a \otimes a', b \otimes b'), \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b') ).

The arguments are a list of four objects [ a, a', b, b' ], and two objects s = \mathrm{\underline{coHom}}(a \otimes a', b \otimes b') and r = \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b'). The output is the natural morphism \mathrm{InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{coHom}}(a \otimes a', b \otimes b') \rightarrow \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b').

##### 1.6-25 CoDualityTensorProductCompatibilityMorphism

`‣ CoDualityTensorProductCompatibilityMorphism`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( (a \otimes b)_{\vee}, a_{\vee} \otimes b_{\vee} ).

The arguments are two objects a,b. The output is the natural morphism \mathrm{CoDualityTensorProductCompatibilityMorphismWithGivenObjects}: (a \otimes b)_{\vee} \rightarrow a_{\vee} \otimes b_{\vee}.

##### 1.6-26 CoDualityTensorProductCompatibilityMorphismWithGivenObjects

`‣ CoDualityTensorProductCompatibilityMorphismWithGivenObjects`( s, a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( (a \otimes b)_{\vee}, a_{\vee} \otimes b_{\vee} ).

The arguments are an object s = (a \otimes b)_{\vee}, two objects a,b, and an object r = a_{\vee} \otimes b_{\vee}. The output is the natural morphism \mathrm{CoDualityTensorProductCompatibilityMorphismWithGivenObjects}_{a,b}: (a \otimes b)_{\vee} \rightarrow a_{\vee} \otimes b_{\vee}.

##### 1.6-27 MorphismFromInternalCoHomToTensorProduct

`‣ MorphismFromInternalCoHomToTensorProduct`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), b_{\vee} \otimes a ).

The arguments are two objects a,b. The output is the natural morphism \mathrm{MorphismFromInternalCoHomToTensorProductWithGivenObjects}_{a,b}: \mathrm{\underline{coHom}}(a,b) \rightarrow b_{\vee} \otimes a.

##### 1.6-28 MorphismFromInternalCoHomToTensorProductWithGivenObjects

`‣ MorphismFromInternalCoHomToTensorProductWithGivenObjects`( s, a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), a \otimes b_{\vee} ).

The arguments are an object s = \mathrm{\underline{coHom}}(a,b), two objects a,b, and an object r = b_{\vee} \otimes a. The output is the natural morphism \mathrm{MorphismFromInternalCoHomToTensorProductWithGivenObjects}_{a,b}: \mathrm{\underline{coHom}}(a,b) \rightarrow a \otimes b_{\vee}.

##### 1.6-29 IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit

`‣ IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a_{\vee}, \mathrm{\underline{coHom}}(1,a)).

The argument is an object a. The output is the isomorphism \mathrm{IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit}_{a}: a_{\vee} \rightarrow \mathrm{\underline{coHom}}(1,a).

##### 1.6-30 IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject

`‣ IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(\mathrm{\underline{coHom}}(1,a), a_{\vee}).

The argument is an object a. The output is the isomorphism \mathrm{IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject}_{a}: \mathrm{\underline{coHom}}(1,a) \rightarrow a_{\vee}.

##### 1.6-31 UniversalPropertyOfCoDual

`‣ UniversalPropertyOfCoDual`( t, a, alpha ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a_{\vee}, t).

The arguments are two objects t,a, and a morphism \alpha: 1 \rightarrow t \otimes a. The output is the morphism a_{\vee} \rightarrow t given by the universal property of a_{\vee}.

##### 1.6-32 CoLambdaIntroduction

`‣ CoLambdaIntroduction`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), 1 ).

The argument is a morphism \alpha: a \rightarrow b. The output is the corresponding morphism \mathrm{\underline{coHom}}(a,b) \rightarrow 1 under the cohom tensor adjunction.

##### 1.6-33 CoLambdaElimination

`‣ CoLambdaElimination`( a, b, alpha ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a,b).

The arguments are two objects a,b, and a morphism \alpha: \mathrm{\underline{coHom}}(a,b) \rightarrow 1. The output is a morphism a \rightarrow b corresponding to \alpha under the cohom tensor adjunction.

##### 1.6-34 IsomorphismFromObjectToInternalCoHom

`‣ IsomorphismFromObjectToInternalCoHom`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a, \mathrm{\underline{coHom}}(a,1)).

The argument is an object a. The output is the natural isomorphism a \rightarrow \mathrm{\underline{coHom}}(a,1).

##### 1.6-35 IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom

`‣ IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom`( a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a, \mathrm{\underline{coHom}}(a,1)).

The argument is an object a, and an object r = \mathrm{\underline{coHom}}(a,1). The output is the natural isomorphism a \rightarrow \mathrm{\underline{coHom}}(a,1).

##### 1.6-36 IsomorphismFromInternalCoHomToObject

`‣ IsomorphismFromInternalCoHomToObject`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(\mathrm{\underline{coHom}}(a,1), a).

The argument is an object a. The output is the natural isomorphism \mathrm{\underline{coHom}}(a,1) \rightarrow a.

##### 1.6-37 IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom

`‣ IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom`( a, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}(\mathrm{\underline{coHom}}(a,1), a).

The argument is an object a, and an object s = \mathrm{\underline{coHom}}(a,1). The output is the natural isomorphism \mathrm{\underline{coHom}}(a,1) \rightarrow a.

#### 1.7 Symmetric Closed Monoidal Categories

A monoidal category \mathbf{C} which is symmetric and closed is called a symmetric closed monoidal category.

The corresponding GAP property is given by `IsSymmetricClosedMonoidalCategory`.

#### 1.8 Symmetric Coclosed Monoidal Categories

A monoidal category \mathbf{C} which is symmetric and coclosed is called a symmetric coclosed monoidal category.

The corresponding GAP property is given by `IsSymmetricCoclosedMonoidalCategory`.

#### 1.9 Rigid Symmetric Closed Monoidal Categories

A symmetric closed monoidal category \mathbf{C} satisfying

- the natural morphism

\mathrm{\underline{Hom}}(a, a') \otimes \mathrm{\underline{Hom}}(b, b') \rightarrow \mathrm{\underline{Hom}}(a \otimes b, a' \otimes b') is an isomorphism,

- the natural morphism

a \rightarrow \mathrm{\underline{Hom}}(\mathrm{\underline{Hom}}(a, 1), 1) is an isomorphism is called a rigid symmetric closed monoidal category.

If no operations involving the closed structure are installed manually, the internal hom objects will be derived as \mathrm{\underline{Hom}}(a,b) \coloneqq a^\vee \otimes b and, in particular, \mathrm{\underline{Hom}}(a,1) \coloneqq a^\vee \otimes 1.

The corresponding GAP property is given by `IsRigidSymmetricClosedMonoidalCategory`.

##### 1.9-1 IsomorphismFromTensorProductWithDualObjectToInternalHom

`‣ IsomorphismFromTensorProductWithDualObjectToInternalHom`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a^{\vee} \otimes b, \mathrm{\underline{Hom}}(a,b) ).

The arguments are two objects a,b. The output is the natural morphism \mathrm{IsomorphismFromTensorProductWithDualObjectToInternalHom}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}}(a,b).

##### 1.9-2 IsomorphismFromInternalHomToTensorProductWithDualObject

`‣ IsomorphismFromInternalHomToTensorProductWithDualObject`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(a,b), a^{\vee} \otimes b ).

The arguments are two objects a,b. The output is the inverse of \mathrm{IsomorphismFromTensorProductWithDualObjectToInternalHom}, namely \mathrm{IsomorphismFromInternalHomToTensorProductWithDualObject}_{a,b}: \mathrm{\underline{Hom}}(a,b) \rightarrow a^{\vee} \otimes b.

##### 1.9-3 MorphismFromInternalHomToTensorProduct

`‣ MorphismFromInternalHomToTensorProduct`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(a,b), a^{\vee} \otimes b ).

The arguments are two objects a,b. The output is the inverse of \mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}, namely \mathrm{MorphismFromInternalHomToTensorProductWithGivenObjects}_{a,b}: \mathrm{\underline{Hom}}(a,b) \rightarrow a^{\vee} \otimes b.

##### 1.9-4 MorphismFromInternalHomToTensorProductWithGivenObjects

`‣ MorphismFromInternalHomToTensorProductWithGivenObjects`( s, a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(a,b), a^{\vee} \otimes b ).

The arguments are an object s = \mathrm{\underline{Hom}}(a,b), two objects a,b, and an object r = a^{\vee} \otimes b. The output is the inverse of \mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}, namely \mathrm{MorphismFromInternalHomToTensorProductWithGivenObjects}_{a,b}: \mathrm{\underline{Hom}}(a,b) \rightarrow a^{\vee} \otimes b.

##### 1.9-5 TensorProductInternalHomCompatibilityMorphismInverse

`‣ TensorProductInternalHomCompatibilityMorphismInverse`( list ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b'), \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b') ).

The argument is a list of four objects [ a, a', b, b' ]. The output is the natural morphism \mathrm{TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b') \rightarrow \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b').

##### 1.9-6 TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects

`‣ TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects`( s, list, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b'), \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b') ).

The arguments are a list of four objects [ a, a', b, b' ], and two objects s = \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b') and r = \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b'). The output is the natural morphism \mathrm{TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b') \rightarrow \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b').

##### 1.9-7 CoevaluationForDual

`‣ CoevaluationForDual`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(1,a \otimes a^{\vee}).

The argument is an object a. The output is the coevaluation morphism \mathrm{coev}_{a}:1 \rightarrow a \otimes a^{\vee}.

##### 1.9-8 CoevaluationForDualWithGivenTensorProduct

`‣ CoevaluationForDualWithGivenTensorProduct`( s, a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}(1,a \otimes a^{\vee}).

The arguments are an object s = 1, an object a, and an object r = a \otimes a^{\vee}. The output is the coevaluation morphism \mathrm{coev}_{a}:1 \rightarrow a \otimes a^{\vee}.

##### 1.9-9 TraceMap

`‣ TraceMap`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(1,1).

The argument is an endomorphism \alpha: a \rightarrow a. The output is the trace morphism \mathrm{trace}_{\alpha}: 1 \rightarrow 1.

##### 1.9-10 RankMorphism

`‣ RankMorphism`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(1,1).

The argument is an object a. The output is the rank morphism \mathrm{rank}_a: 1 \rightarrow 1.

##### 1.9-11 MorphismFromBidual

`‣ MorphismFromBidual`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}((a^{\vee})^{\vee},a).

The argument is an object a. The output is the inverse of the morphism to the bidual (a^{\vee})^{\vee} \rightarrow a.

##### 1.9-12 MorphismFromBidualWithGivenBidual

`‣ MorphismFromBidualWithGivenBidual`( a, s ) | ( operation )

Returns: a morphism in \mathrm{Hom}((a^{\vee})^{\vee},a).

The argument is an object a, and an object s = (a^{\vee})^{\vee}. The output is the inverse of the morphism to the bidual (a^{\vee})^{\vee} \rightarrow a.

#### 1.10 Rigid Symmetric Coclosed Monoidal Categories

A symmetric coclosed monoidal category \mathbf{C} satisfying

- the natural morphism

\mathrm{\underline{coHom}}(a \otimes a', b \otimes b') \rightarrow \mathrm{\underline{coHom}}(a, b) \otimes \mathrm{\underline{coHom}}(a', b') is an isomorphism,

- the natural morphism

\mathrm{\underline{coHom}}(1, \mathrm{\underline{coHom}}(1, a)) \rightarrow a is an isomorphism is called a rigid symmetric coclosed monoidal category.

If no operations involving the coclosed structure are installed manually, the internal cohom objects will be derived as \mathrm{\underline{coHom}}(a,b) \coloneqq a \otimes b_\vee and, in particular, \mathrm{\underline{coHom}}(1,a) \coloneqq 1 \otimes a_\vee.

The corresponding GAP property is given by `IsRigidSymmetricCoclosedMonoidalCategory`.

##### 1.10-1 IsomorphismFromInternalCoHomToTensorProductWithCoDualObject

`‣ IsomorphismFromInternalCoHomToTensorProductWithCoDualObject`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b), b_{\vee} \otimes a ).

The arguments are two objects a,b. The output is the natural morphism \mathrm{IsomorphismFromInternalCoHomToTensorProductWithCoDualObjectWithGivenObjects}_{a,b}: \mathrm{\underline{coHom}}(a,b) \rightarrow b_{\vee} \otimes a.

##### 1.10-2 IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom

`‣ IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a_{\vee} \otimes b, \mathrm{\underline{coHom}}(b,a).

The arguments are two objects a,b. The output is the inverse of \mathrm{IsomorphismFromInternalCoHomToTensorProductWithCoDualObject}, namely \mathrm{IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom}_{a,b}: a_{\vee} \otimes b \rightarrow \mathrm{\underline{coHom}}(b,a).

##### 1.10-3 MorphismFromTensorProductToInternalCoHom

`‣ MorphismFromTensorProductToInternalCoHom`( a, b ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a_{\vee} \otimes b, \mathrm{\underline{coHom}}(b,a) ).

The arguments are two objects a,b. The output is the inverse of \mathrm{MorphismFromInternalCoHomToTensorProductWithGivenObjects}, namely \mathrm{MorphismFromTensorProductToInternalCoHomWithGivenObjects}_{a,b}: a_{\vee} \otimes b \rightarrow \mathrm{\underline{coHom}}(b,a).

##### 1.10-4 MorphismFromTensorProductToInternalCoHomWithGivenObjects

`‣ MorphismFromTensorProductToInternalCoHomWithGivenObjects`( s, a, b, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( a_{\vee} \otimes b, \mathrm{\underline{coHom}}(b,a).

The arguments are an object s_{\vee} = a \otimes b, two objects a,b, and an object r = \mathrm{\underline{coHom}}(b,a). The output is the inverse of \mathrm{MorphismFromInternalCoHomToTensorProductWithGivenObjects}, namely \mathrm{MorphismFromTensorProductToInternalCoHomWithGivenObjects}_{a,b}: a_{\vee} \otimes b \rightarrow \mathrm{\underline{coHom}}(b,a).

##### 1.10-5 InternalCoHomTensorProductCompatibilityMorphismInverse

`‣ InternalCoHomTensorProductCompatibilityMorphismInverse`( list ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b'), \mathrm{\underline{coHom}}(a \otimes a', b \otimes b' ).

The argument is a list of four objects [ a, a', b, b' ]. The output is the natural morphism \mathrm{InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b') \rightarrow \mathrm{\underline{coHom}}(a \otimes a', b \otimes b').

##### 1.10-6 InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects

`‣ InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects`( s, list, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b'), \mathrm{\underline{coHom}}(a \otimes a', b \otimes b' ).

The arguments are a list of four objects [ a, a', b, b' ], and two objects s = \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b') and r = \mathrm{\underline{coHom}}(a \otimes a', b \otimes b'). The output is the natural morphism \mathrm{InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{coHom}}(a,b) \otimes \mathrm{\underline{coHom}}(a',b') \rightarrow \mathrm{\underline{coHom}}(a \otimes a', b \otimes b').

##### 1.10-7 CoclosedCoevaluationForCoDual

`‣ CoclosedCoevaluationForCoDual`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a \otimes a_{\vee}, 1).

The argument is an object a. The output is the coclosed coevaluation morphism \mathrm{coclcoev}_{a}: a \otimes a_{\vee} \rightarrow 1.

##### 1.10-8 CoclosedCoevaluationForCoDualWithGivenTensorProduct

`‣ CoclosedCoevaluationForCoDualWithGivenTensorProduct`( s, a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a \otimes a_{\vee}, 1).

The arguments are an object s = a \otimes a_{\vee}, an object a, and an object r = 1. The output is the coclosed coevaluation morphism \mathrm{coclcoev}_{a}: a \otimes a_{\vee} \rightarrow 1.

##### 1.10-9 CoTraceMap

`‣ CoTraceMap`( alpha ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(1,1).

The argument is an endomorphism \alpha: a \rightarrow a. The output is the cotrace morphism \mathrm{cotrace}_{\alpha}: 1 \rightarrow 1.

##### 1.10-10 CoRankMorphism

`‣ CoRankMorphism`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(1,1).

The argument is an object a. The output is the corank morphism \mathrm{corank}_a: 1 \rightarrow 1.

##### 1.10-11 MorphismToCoBidual

`‣ MorphismToCoBidual`( a ) | ( attribute )

Returns: a morphism in \mathrm{Hom}(a, (a_{\vee})_{\vee}).

The argument is an object a. The output is the inverse of the morphism from the cobidual a \rightarrow (a_{\vee})_{\vee}.

##### 1.10-12 MorphismToCoBidualWithGivenCoBidual

`‣ MorphismToCoBidualWithGivenCoBidual`( a, r ) | ( operation )

Returns: a morphism in \mathrm{Hom}(a,(a_{\vee})_{\vee}).

The argument is an object a, and an object r = (a_{\vee})_{\vee}. The output is the inverse of the morphism from the cobidual a \rightarrow (a_{\vee})_{\vee}.

#### 1.11 Convenience Methods

##### 1.11-1 InternalHom

`‣ InternalHom`( a, b ) | ( operation )

Returns: a cell

This is a convenience method. The arguments are two cells a,b. The output is the internal hom cell. If a,b are two CAP objects the output is the internal Hom object \mathrm{\underline{Hom}}(a,b). If at least one of the arguments is a CAP morphism the output is a CAP morphism, namely the internal hom on morphisms, where any object is replaced by its identity morphism.

##### 1.11-2 InternalCoHom

`‣ InternalCoHom`( a, b ) | ( operation )

Returns: a cell

This is a convenience method. The arguments are two cells a,b. The output is the internal cohom cell. If a,b are two CAP objects the output is the internal cohom object \mathrm{\underline{coHom}}(a,b). If at least one of the arguments is a CAP morphism the output is a CAP morphism, namely the internal cohom on morphisms, where any object is replaced by its identity morphism.

#### 1.12 Add-methods

##### 1.12-1 AddLeftDistributivityExpanding

`‣ AddLeftDistributivityExpanding`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftDistributivityExpanding`. F: ( a, L ) \mapsto \mathtt{LeftDistributivityExpanding}(a, L).

##### 1.12-2 AddLeftDistributivityExpandingWithGivenObjects

`‣ AddLeftDistributivityExpandingWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftDistributivityExpandingWithGivenObjects`. F: ( s, a, L, r ) \mapsto \mathtt{LeftDistributivityExpandingWithGivenObjects}(s, a, L, r).

##### 1.12-3 AddLeftDistributivityFactoring

`‣ AddLeftDistributivityFactoring`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftDistributivityFactoring`. F: ( a, L ) \mapsto \mathtt{LeftDistributivityFactoring}(a, L).

##### 1.12-4 AddLeftDistributivityFactoringWithGivenObjects

`‣ AddLeftDistributivityFactoringWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftDistributivityFactoringWithGivenObjects`. F: ( s, a, L, r ) \mapsto \mathtt{LeftDistributivityFactoringWithGivenObjects}(s, a, L, r).

##### 1.12-5 AddRightDistributivityExpanding

`‣ AddRightDistributivityExpanding`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `RightDistributivityExpanding`. F: ( L, a ) \mapsto \mathtt{RightDistributivityExpanding}(L, a).

##### 1.12-6 AddRightDistributivityExpandingWithGivenObjects

`‣ AddRightDistributivityExpandingWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `RightDistributivityExpandingWithGivenObjects`. F: ( s, L, a, r ) \mapsto \mathtt{RightDistributivityExpandingWithGivenObjects}(s, L, a, r).

##### 1.12-7 AddRightDistributivityFactoring

`‣ AddRightDistributivityFactoring`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `RightDistributivityFactoring`. F: ( L, a ) \mapsto \mathtt{RightDistributivityFactoring}(L, a).

##### 1.12-8 AddRightDistributivityFactoringWithGivenObjects

`‣ AddRightDistributivityFactoringWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `RightDistributivityFactoringWithGivenObjects`. F: ( s, L, a, r ) \mapsto \mathtt{RightDistributivityFactoringWithGivenObjects}(s, L, a, r).

##### 1.12-9 AddBraiding

`‣ AddBraiding`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `Braiding`. F: ( a, b ) \mapsto \mathtt{Braiding}(a, b).

##### 1.12-10 AddBraidingInverse

`‣ AddBraidingInverse`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `BraidingInverse`. F: ( a, b ) \mapsto \mathtt{BraidingInverse}(a, b).

##### 1.12-11 AddBraidingInverseWithGivenTensorProducts

`‣ AddBraidingInverseWithGivenTensorProducts`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `BraidingInverseWithGivenTensorProducts`. F: ( s, a, b, r ) \mapsto \mathtt{BraidingInverseWithGivenTensorProducts}(s, a, b, r).

##### 1.12-12 AddBraidingWithGivenTensorProducts

`‣ AddBraidingWithGivenTensorProducts`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `BraidingWithGivenTensorProducts`. F: ( s, a, b, r ) \mapsto \mathtt{BraidingWithGivenTensorProducts}(s, a, b, r).

##### 1.12-13 AddCoevaluationMorphism

`‣ AddCoevaluationMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoevaluationMorphism`. F: ( a, b ) \mapsto \mathtt{CoevaluationMorphism}(a, b).

##### 1.12-14 AddCoevaluationMorphismWithGivenRange

`‣ AddCoevaluationMorphismWithGivenRange`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoevaluationMorphismWithGivenRange`. F: ( a, b, r ) \mapsto \mathtt{CoevaluationMorphismWithGivenRange}(a, b, r).

##### 1.12-15 AddDualOnMorphisms

`‣ AddDualOnMorphisms`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `DualOnMorphisms`. F: ( alpha ) \mapsto \mathtt{DualOnMorphisms}(alpha).

##### 1.12-16 AddDualOnMorphismsWithGivenDuals

`‣ AddDualOnMorphismsWithGivenDuals`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `DualOnMorphismsWithGivenDuals`. F: ( s, alpha, r ) \mapsto \mathtt{DualOnMorphismsWithGivenDuals}(s, alpha, r).

##### 1.12-17 AddDualOnObjects

`‣ AddDualOnObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `DualOnObjects`. F: ( a ) \mapsto \mathtt{DualOnObjects}(a).

##### 1.12-18 AddEvaluationForDual

`‣ AddEvaluationForDual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `EvaluationForDual`. F: ( a ) \mapsto \mathtt{EvaluationForDual}(a).

##### 1.12-19 AddEvaluationForDualWithGivenTensorProduct

`‣ AddEvaluationForDualWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `EvaluationForDualWithGivenTensorProduct`. F: ( s, a, r ) \mapsto \mathtt{EvaluationForDualWithGivenTensorProduct}(s, a, r).

##### 1.12-20 AddEvaluationMorphism

`‣ AddEvaluationMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `EvaluationMorphism`. F: ( a, b ) \mapsto \mathtt{EvaluationMorphism}(a, b).

##### 1.12-21 AddEvaluationMorphismWithGivenSource

`‣ AddEvaluationMorphismWithGivenSource`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `EvaluationMorphismWithGivenSource`. F: ( a, b, s ) \mapsto \mathtt{EvaluationMorphismWithGivenSource}(a, b, s).

##### 1.12-22 AddInternalHomOnMorphisms

`‣ AddInternalHomOnMorphisms`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalHomOnMorphisms`. F: ( alpha, beta ) \mapsto \mathtt{InternalHomOnMorphisms}(alpha, beta).

##### 1.12-23 AddInternalHomOnMorphismsWithGivenInternalHoms

`‣ AddInternalHomOnMorphismsWithGivenInternalHoms`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalHomOnMorphismsWithGivenInternalHoms`. F: ( s, alpha, beta, r ) \mapsto \mathtt{InternalHomOnMorphismsWithGivenInternalHoms}(s, alpha, beta, r).

##### 1.12-24 AddInternalHomOnObjects

`‣ AddInternalHomOnObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalHomOnObjects`. F: ( a, b ) \mapsto \mathtt{InternalHomOnObjects}(a, b).

##### 1.12-25 AddInternalHomToTensorProductAdjunctionMap

`‣ AddInternalHomToTensorProductAdjunctionMap`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalHomToTensorProductAdjunctionMap`. F: ( b, c, g ) \mapsto \mathtt{InternalHomToTensorProductAdjunctionMap}(b, c, g).

##### 1.12-26 AddInternalHomToTensorProductAdjunctionMapWithGivenTensorProduct

`‣ AddInternalHomToTensorProductAdjunctionMapWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalHomToTensorProductAdjunctionMapWithGivenTensorProduct`. F: ( b, c, g, t ) \mapsto \mathtt{InternalHomToTensorProductAdjunctionMapWithGivenTensorProduct}(b, c, g, t).

##### 1.12-27 AddIsomorphismFromDualObjectToInternalHomIntoTensorUnit

`‣ AddIsomorphismFromDualObjectToInternalHomIntoTensorUnit`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromDualObjectToInternalHomIntoTensorUnit`. F: ( a ) \mapsto \mathtt{IsomorphismFromDualObjectToInternalHomIntoTensorUnit}(a).

##### 1.12-28 AddIsomorphismFromInternalHomIntoTensorUnitToDualObject

`‣ AddIsomorphismFromInternalHomIntoTensorUnitToDualObject`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromInternalHomIntoTensorUnitToDualObject`. F: ( a ) \mapsto \mathtt{IsomorphismFromInternalHomIntoTensorUnitToDualObject}(a).

##### 1.12-29 AddIsomorphismFromInternalHomToObject

`‣ AddIsomorphismFromInternalHomToObject`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromInternalHomToObject`. F: ( a ) \mapsto \mathtt{IsomorphismFromInternalHomToObject}(a).

##### 1.12-30 AddIsomorphismFromInternalHomToObjectWithGivenInternalHom

`‣ AddIsomorphismFromInternalHomToObjectWithGivenInternalHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromInternalHomToObjectWithGivenInternalHom`. F: ( a, s ) \mapsto \mathtt{IsomorphismFromInternalHomToObjectWithGivenInternalHom}(a, s).

##### 1.12-31 AddIsomorphismFromObjectToInternalHom

`‣ AddIsomorphismFromObjectToInternalHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromObjectToInternalHom`. F: ( a ) \mapsto \mathtt{IsomorphismFromObjectToInternalHom}(a).

##### 1.12-32 AddIsomorphismFromObjectToInternalHomWithGivenInternalHom

`‣ AddIsomorphismFromObjectToInternalHomWithGivenInternalHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromObjectToInternalHomWithGivenInternalHom`. F: ( a, r ) \mapsto \mathtt{IsomorphismFromObjectToInternalHomWithGivenInternalHom}(a, r).

##### 1.12-33 AddLambdaElimination

`‣ AddLambdaElimination`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LambdaElimination`. F: ( a, b, alpha ) \mapsto \mathtt{LambdaElimination}(a, b, alpha).

##### 1.12-34 AddLambdaIntroduction

`‣ AddLambdaIntroduction`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LambdaIntroduction`. F: ( alpha ) \mapsto \mathtt{LambdaIntroduction}(alpha).

##### 1.12-35 AddMonoidalPostComposeMorphism

`‣ AddMonoidalPostComposeMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MonoidalPostComposeMorphism`. F: ( a, b, c ) \mapsto \mathtt{MonoidalPostComposeMorphism}(a, b, c).

##### 1.12-36 AddMonoidalPostComposeMorphismWithGivenObjects

`‣ AddMonoidalPostComposeMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MonoidalPostComposeMorphismWithGivenObjects`. F: ( s, a, b, c, r ) \mapsto \mathtt{MonoidalPostComposeMorphismWithGivenObjects}(s, a, b, c, r).

##### 1.12-37 AddMonoidalPreComposeMorphism

`‣ AddMonoidalPreComposeMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MonoidalPreComposeMorphism`. F: ( a, b, c ) \mapsto \mathtt{MonoidalPreComposeMorphism}(a, b, c).

##### 1.12-38 AddMonoidalPreComposeMorphismWithGivenObjects

`‣ AddMonoidalPreComposeMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MonoidalPreComposeMorphismWithGivenObjects`. F: ( s, a, b, c, r ) \mapsto \mathtt{MonoidalPreComposeMorphismWithGivenObjects}(s, a, b, c, r).

##### 1.12-39 AddMorphismFromTensorProductToInternalHom

`‣ AddMorphismFromTensorProductToInternalHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromTensorProductToInternalHom`. F: ( a, b ) \mapsto \mathtt{MorphismFromTensorProductToInternalHom}(a, b).

##### 1.12-40 AddMorphismFromTensorProductToInternalHomWithGivenObjects

`‣ AddMorphismFromTensorProductToInternalHomWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromTensorProductToInternalHomWithGivenObjects`. F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromTensorProductToInternalHomWithGivenObjects}(s, a, b, r).

##### 1.12-41 AddMorphismToBidual

`‣ AddMorphismToBidual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismToBidual`. F: ( a ) \mapsto \mathtt{MorphismToBidual}(a).

##### 1.12-42 AddMorphismToBidualWithGivenBidual

`‣ AddMorphismToBidualWithGivenBidual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismToBidualWithGivenBidual`. F: ( a, r ) \mapsto \mathtt{MorphismToBidualWithGivenBidual}(a, r).

##### 1.12-43 AddTensorProductDualityCompatibilityMorphism

`‣ AddTensorProductDualityCompatibilityMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductDualityCompatibilityMorphism`. F: ( a, b ) \mapsto \mathtt{TensorProductDualityCompatibilityMorphism}(a, b).

##### 1.12-44 AddTensorProductDualityCompatibilityMorphismWithGivenObjects

`‣ AddTensorProductDualityCompatibilityMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductDualityCompatibilityMorphismWithGivenObjects`. F: ( s, a, b, r ) \mapsto \mathtt{TensorProductDualityCompatibilityMorphismWithGivenObjects}(s, a, b, r).

##### 1.12-45 AddTensorProductInternalHomCompatibilityMorphism

`‣ AddTensorProductInternalHomCompatibilityMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductInternalHomCompatibilityMorphism`. F: ( list ) \mapsto \mathtt{TensorProductInternalHomCompatibilityMorphism}(list).

##### 1.12-46 AddTensorProductInternalHomCompatibilityMorphismWithGivenObjects

`‣ AddTensorProductInternalHomCompatibilityMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductInternalHomCompatibilityMorphismWithGivenObjects`. F: ( source, list, range ) \mapsto \mathtt{TensorProductInternalHomCompatibilityMorphismWithGivenObjects}(source, list, range).

##### 1.12-47 AddTensorProductToInternalHomAdjunctionMap

`‣ AddTensorProductToInternalHomAdjunctionMap`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductToInternalHomAdjunctionMap`. F: ( a, b, f ) \mapsto \mathtt{TensorProductToInternalHomAdjunctionMap}(a, b, f).

##### 1.12-48 AddTensorProductToInternalHomAdjunctionMapWithGivenInternalHom

`‣ AddTensorProductToInternalHomAdjunctionMapWithGivenInternalHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductToInternalHomAdjunctionMapWithGivenInternalHom`. F: ( a, b, f, i ) \mapsto \mathtt{TensorProductToInternalHomAdjunctionMapWithGivenInternalHom}(a, b, f, i).

##### 1.12-49 AddUniversalPropertyOfDual

`‣ AddUniversalPropertyOfDual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `UniversalPropertyOfDual`. F: ( t, a, alpha ) \mapsto \mathtt{UniversalPropertyOfDual}(t, a, alpha).

##### 1.12-50 AddCoDualOnMorphisms

`‣ AddCoDualOnMorphisms`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoDualOnMorphisms`. F: ( alpha ) \mapsto \mathtt{CoDualOnMorphisms}(alpha).

##### 1.12-51 AddCoDualOnMorphismsWithGivenCoDuals

`‣ AddCoDualOnMorphismsWithGivenCoDuals`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoDualOnMorphismsWithGivenCoDuals`. F: ( s, alpha, r ) \mapsto \mathtt{CoDualOnMorphismsWithGivenCoDuals}(s, alpha, r).

##### 1.12-52 AddCoDualOnObjects

`‣ AddCoDualOnObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoDualOnObjects`. F: ( a ) \mapsto \mathtt{CoDualOnObjects}(a).

##### 1.12-53 AddCoDualityTensorProductCompatibilityMorphism

`‣ AddCoDualityTensorProductCompatibilityMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoDualityTensorProductCompatibilityMorphism`. F: ( a, b ) \mapsto \mathtt{CoDualityTensorProductCompatibilityMorphism}(a, b).

##### 1.12-54 AddCoDualityTensorProductCompatibilityMorphismWithGivenObjects

`‣ AddCoDualityTensorProductCompatibilityMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoDualityTensorProductCompatibilityMorphismWithGivenObjects`. F: ( s, a, b, r ) \mapsto \mathtt{CoDualityTensorProductCompatibilityMorphismWithGivenObjects}(s, a, b, r).

##### 1.12-55 AddCoLambdaElimination

`‣ AddCoLambdaElimination`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoLambdaElimination`. F: ( a, b, alpha ) \mapsto \mathtt{CoLambdaElimination}(a, b, alpha).

##### 1.12-56 AddCoLambdaIntroduction

`‣ AddCoLambdaIntroduction`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoLambdaIntroduction`. F: ( alpha ) \mapsto \mathtt{CoLambdaIntroduction}(alpha).

##### 1.12-57 AddCoclosedCoevaluationMorphism

`‣ AddCoclosedCoevaluationMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoclosedCoevaluationMorphism`. F: ( a, b ) \mapsto \mathtt{CoclosedCoevaluationMorphism}(a, b).

##### 1.12-58 AddCoclosedCoevaluationMorphismWithGivenSource

`‣ AddCoclosedCoevaluationMorphismWithGivenSource`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoclosedCoevaluationMorphismWithGivenSource`. F: ( a, b, s ) \mapsto \mathtt{CoclosedCoevaluationMorphismWithGivenSource}(a, b, s).

##### 1.12-59 AddCoclosedEvaluationForCoDual

`‣ AddCoclosedEvaluationForCoDual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoclosedEvaluationForCoDual`. F: ( a ) \mapsto \mathtt{CoclosedEvaluationForCoDual}(a).

##### 1.12-60 AddCoclosedEvaluationForCoDualWithGivenTensorProduct

`‣ AddCoclosedEvaluationForCoDualWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoclosedEvaluationForCoDualWithGivenTensorProduct`. F: ( s, a, r ) \mapsto \mathtt{CoclosedEvaluationForCoDualWithGivenTensorProduct}(s, a, r).

##### 1.12-61 AddCoclosedEvaluationMorphism

`‣ AddCoclosedEvaluationMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoclosedEvaluationMorphism`. F: ( a, b ) \mapsto \mathtt{CoclosedEvaluationMorphism}(a, b).

##### 1.12-62 AddCoclosedEvaluationMorphismWithGivenRange

`‣ AddCoclosedEvaluationMorphismWithGivenRange`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoclosedEvaluationMorphismWithGivenRange`. F: ( a, b, r ) \mapsto \mathtt{CoclosedEvaluationMorphismWithGivenRange}(a, b, r).

##### 1.12-63 AddInternalCoHomOnMorphisms

`‣ AddInternalCoHomOnMorphisms`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalCoHomOnMorphisms`. F: ( alpha, beta ) \mapsto \mathtt{InternalCoHomOnMorphisms}(alpha, beta).

##### 1.12-64 AddInternalCoHomOnMorphismsWithGivenInternalCoHoms

`‣ AddInternalCoHomOnMorphismsWithGivenInternalCoHoms`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalCoHomOnMorphismsWithGivenInternalCoHoms`. F: ( s, alpha, beta, r ) \mapsto \mathtt{InternalCoHomOnMorphismsWithGivenInternalCoHoms}(s, alpha, beta, r).

##### 1.12-65 AddInternalCoHomOnObjects

`‣ AddInternalCoHomOnObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalCoHomOnObjects`. F: ( a, b ) \mapsto \mathtt{InternalCoHomOnObjects}(a, b).

##### 1.12-66 AddInternalCoHomTensorProductCompatibilityMorphism

`‣ AddInternalCoHomTensorProductCompatibilityMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalCoHomTensorProductCompatibilityMorphism`. F: ( list ) \mapsto \mathtt{InternalCoHomTensorProductCompatibilityMorphism}(list).

##### 1.12-67 AddInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects

`‣ AddInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects`. F: ( source, list, range ) \mapsto \mathtt{InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects}(source, list, range).

##### 1.12-68 AddInternalCoHomToTensorProductAdjunctionMap

`‣ AddInternalCoHomToTensorProductAdjunctionMap`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalCoHomToTensorProductAdjunctionMap`. F: ( a, b, f ) \mapsto \mathtt{InternalCoHomToTensorProductAdjunctionMap}(a, b, f).

##### 1.12-69 AddInternalCoHomToTensorProductAdjunctionMapWithGivenTensorProduct

`‣ AddInternalCoHomToTensorProductAdjunctionMapWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalCoHomToTensorProductAdjunctionMapWithGivenTensorProduct`. F: ( a, b, f, t ) \mapsto \mathtt{InternalCoHomToTensorProductAdjunctionMapWithGivenTensorProduct}(a, b, f, t).

##### 1.12-70 AddIsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit

`‣ AddIsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit`. F: ( a ) \mapsto \mathtt{IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit}(a).

##### 1.12-71 AddIsomorphismFromInternalCoHomFromTensorUnitToCoDualObject

`‣ AddIsomorphismFromInternalCoHomFromTensorUnitToCoDualObject`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject`. F: ( a ) \mapsto \mathtt{IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject}(a).

##### 1.12-72 AddIsomorphismFromInternalCoHomToObject

`‣ AddIsomorphismFromInternalCoHomToObject`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromInternalCoHomToObject`. F: ( a ) \mapsto \mathtt{IsomorphismFromInternalCoHomToObject}(a).

##### 1.12-73 AddIsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom

`‣ AddIsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom`. F: ( a, s ) \mapsto \mathtt{IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom}(a, s).

##### 1.12-74 AddIsomorphismFromObjectToInternalCoHom

`‣ AddIsomorphismFromObjectToInternalCoHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromObjectToInternalCoHom`. F: ( a ) \mapsto \mathtt{IsomorphismFromObjectToInternalCoHom}(a).

##### 1.12-75 AddIsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom

`‣ AddIsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom`. F: ( a, r ) \mapsto \mathtt{IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom}(a, r).

##### 1.12-76 AddMonoidalPostCoComposeMorphism

`‣ AddMonoidalPostCoComposeMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MonoidalPostCoComposeMorphism`. F: ( a, b, c ) \mapsto \mathtt{MonoidalPostCoComposeMorphism}(a, b, c).

##### 1.12-77 AddMonoidalPostCoComposeMorphismWithGivenObjects

`‣ AddMonoidalPostCoComposeMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MonoidalPostCoComposeMorphismWithGivenObjects`. F: ( s, a, b, c, r ) \mapsto \mathtt{MonoidalPostCoComposeMorphismWithGivenObjects}(s, a, b, c, r).

##### 1.12-78 AddMonoidalPreCoComposeMorphism

`‣ AddMonoidalPreCoComposeMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MonoidalPreCoComposeMorphism`. F: ( a, b, c ) \mapsto \mathtt{MonoidalPreCoComposeMorphism}(a, b, c).

##### 1.12-79 AddMonoidalPreCoComposeMorphismWithGivenObjects

`‣ AddMonoidalPreCoComposeMorphismWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MonoidalPreCoComposeMorphismWithGivenObjects`. F: ( s, a, b, c, r ) \mapsto \mathtt{MonoidalPreCoComposeMorphismWithGivenObjects}(s, a, b, c, r).

##### 1.12-80 AddMorphismFromCoBidual

`‣ AddMorphismFromCoBidual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromCoBidual`. F: ( a ) \mapsto \mathtt{MorphismFromCoBidual}(a).

##### 1.12-81 AddMorphismFromCoBidualWithGivenCoBidual

`‣ AddMorphismFromCoBidualWithGivenCoBidual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromCoBidualWithGivenCoBidual`. F: ( a, s ) \mapsto \mathtt{MorphismFromCoBidualWithGivenCoBidual}(a, s).

##### 1.12-82 AddMorphismFromInternalCoHomToTensorProduct

`‣ AddMorphismFromInternalCoHomToTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromInternalCoHomToTensorProduct`. F: ( a, b ) \mapsto \mathtt{MorphismFromInternalCoHomToTensorProduct}(a, b).

##### 1.12-83 AddMorphismFromInternalCoHomToTensorProductWithGivenObjects

`‣ AddMorphismFromInternalCoHomToTensorProductWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromInternalCoHomToTensorProductWithGivenObjects`. F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromInternalCoHomToTensorProductWithGivenObjects}(s, a, b, r).

##### 1.12-84 AddTensorProductToInternalCoHomAdjunctionMap

`‣ AddTensorProductToInternalCoHomAdjunctionMap`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductToInternalCoHomAdjunctionMap`. F: ( c, b, g ) \mapsto \mathtt{TensorProductToInternalCoHomAdjunctionMap}(c, b, g).

##### 1.12-85 AddTensorProductToInternalCoHomAdjunctionMapWithGivenInternalCoHom

`‣ AddTensorProductToInternalCoHomAdjunctionMapWithGivenInternalCoHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductToInternalCoHomAdjunctionMapWithGivenInternalCoHom`. F: ( c, b, g, i ) \mapsto \mathtt{TensorProductToInternalCoHomAdjunctionMapWithGivenInternalCoHom}(c, b, g, i).

##### 1.12-86 AddUniversalPropertyOfCoDual

`‣ AddUniversalPropertyOfCoDual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `UniversalPropertyOfCoDual`. F: ( t, a, alpha ) \mapsto \mathtt{UniversalPropertyOfCoDual}(t, a, alpha).

##### 1.12-87 AddAssociatorLeftToRight

`‣ AddAssociatorLeftToRight`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `AssociatorLeftToRight`. F: ( a, b, c ) \mapsto \mathtt{AssociatorLeftToRight}(a, b, c).

##### 1.12-88 AddAssociatorLeftToRightWithGivenTensorProducts

`‣ AddAssociatorLeftToRightWithGivenTensorProducts`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `AssociatorLeftToRightWithGivenTensorProducts`. F: ( s, a, b, c, r ) \mapsto \mathtt{AssociatorLeftToRightWithGivenTensorProducts}(s, a, b, c, r).

##### 1.12-89 AddAssociatorRightToLeft

`‣ AddAssociatorRightToLeft`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `AssociatorRightToLeft`. F: ( a, b, c ) \mapsto \mathtt{AssociatorRightToLeft}(a, b, c).

##### 1.12-90 AddAssociatorRightToLeftWithGivenTensorProducts

`‣ AddAssociatorRightToLeftWithGivenTensorProducts`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `AssociatorRightToLeftWithGivenTensorProducts`. F: ( s, a, b, c, r ) \mapsto \mathtt{AssociatorRightToLeftWithGivenTensorProducts}(s, a, b, c, r).

##### 1.12-91 AddLeftUnitor

`‣ AddLeftUnitor`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftUnitor`. F: ( a ) \mapsto \mathtt{LeftUnitor}(a).

##### 1.12-92 AddLeftUnitorInverse

`‣ AddLeftUnitorInverse`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftUnitorInverse`. F: ( a ) \mapsto \mathtt{LeftUnitorInverse}(a).

##### 1.12-93 AddLeftUnitorInverseWithGivenTensorProduct

`‣ AddLeftUnitorInverseWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftUnitorInverseWithGivenTensorProduct`. F: ( a, r ) \mapsto \mathtt{LeftUnitorInverseWithGivenTensorProduct}(a, r).

##### 1.12-94 AddLeftUnitorWithGivenTensorProduct

`‣ AddLeftUnitorWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `LeftUnitorWithGivenTensorProduct`. F: ( a, s ) \mapsto \mathtt{LeftUnitorWithGivenTensorProduct}(a, s).

##### 1.12-95 AddRightUnitor

`‣ AddRightUnitor`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `RightUnitor`. F: ( a ) \mapsto \mathtt{RightUnitor}(a).

##### 1.12-96 AddRightUnitorInverse

`‣ AddRightUnitorInverse`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `RightUnitorInverse`. F: ( a ) \mapsto \mathtt{RightUnitorInverse}(a).

##### 1.12-97 AddRightUnitorInverseWithGivenTensorProduct

`‣ AddRightUnitorInverseWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `RightUnitorInverseWithGivenTensorProduct`. F: ( a, r ) \mapsto \mathtt{RightUnitorInverseWithGivenTensorProduct}(a, r).

##### 1.12-98 AddRightUnitorWithGivenTensorProduct

`‣ AddRightUnitorWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `RightUnitorWithGivenTensorProduct`. F: ( a, s ) \mapsto \mathtt{RightUnitorWithGivenTensorProduct}(a, s).

##### 1.12-99 AddTensorProductOnMorphisms

`‣ AddTensorProductOnMorphisms`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductOnMorphisms`. F: ( alpha, beta ) \mapsto \mathtt{TensorProductOnMorphisms}(alpha, beta).

##### 1.12-100 AddTensorProductOnMorphismsWithGivenTensorProducts

`‣ AddTensorProductOnMorphismsWithGivenTensorProducts`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductOnMorphismsWithGivenTensorProducts`. F: ( s, alpha, beta, r ) \mapsto \mathtt{TensorProductOnMorphismsWithGivenTensorProducts}(s, alpha, beta, r).

##### 1.12-101 AddCoevaluationForDual

`‣ AddCoevaluationForDual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoevaluationForDual`. F: ( a ) \mapsto \mathtt{CoevaluationForDual}(a).

##### 1.12-102 AddCoevaluationForDualWithGivenTensorProduct

`‣ AddCoevaluationForDualWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoevaluationForDualWithGivenTensorProduct`. F: ( s, a, r ) \mapsto \mathtt{CoevaluationForDualWithGivenTensorProduct}(s, a, r).

##### 1.12-103 AddIsomorphismFromInternalHomToTensorProductWithDualObject

`‣ AddIsomorphismFromInternalHomToTensorProductWithDualObject`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromInternalHomToTensorProductWithDualObject`. F: ( a, b ) \mapsto \mathtt{IsomorphismFromInternalHomToTensorProductWithDualObject}(a, b).

##### 1.12-104 AddIsomorphismFromTensorProductWithDualObjectToInternalHom

`‣ AddIsomorphismFromTensorProductWithDualObjectToInternalHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromTensorProductWithDualObjectToInternalHom`. F: ( a, b ) \mapsto \mathtt{IsomorphismFromTensorProductWithDualObjectToInternalHom}(a, b).

##### 1.12-105 AddMorphismFromBidual

`‣ AddMorphismFromBidual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromBidual`. F: ( a ) \mapsto \mathtt{MorphismFromBidual}(a).

##### 1.12-106 AddMorphismFromBidualWithGivenBidual

`‣ AddMorphismFromBidualWithGivenBidual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromBidualWithGivenBidual`. F: ( a, s ) \mapsto \mathtt{MorphismFromBidualWithGivenBidual}(a, s).

##### 1.12-107 AddMorphismFromInternalHomToTensorProduct

`‣ AddMorphismFromInternalHomToTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromInternalHomToTensorProduct`. F: ( a, b ) \mapsto \mathtt{MorphismFromInternalHomToTensorProduct}(a, b).

##### 1.12-108 AddMorphismFromInternalHomToTensorProductWithGivenObjects

`‣ AddMorphismFromInternalHomToTensorProductWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromInternalHomToTensorProductWithGivenObjects`. F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromInternalHomToTensorProductWithGivenObjects}(s, a, b, r).

##### 1.12-109 AddRankMorphism

`‣ AddRankMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `RankMorphism`. F: ( a ) \mapsto \mathtt{RankMorphism}(a).

##### 1.12-110 AddTensorProductInternalHomCompatibilityMorphismInverse

`‣ AddTensorProductInternalHomCompatibilityMorphismInverse`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductInternalHomCompatibilityMorphismInverse`. F: ( list ) \mapsto \mathtt{TensorProductInternalHomCompatibilityMorphismInverse}(list).

##### 1.12-111 AddTensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects

`‣ AddTensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects`. F: ( source, list, range ) \mapsto \mathtt{TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects}(source, list, range).

##### 1.12-112 AddTraceMap

`‣ AddTraceMap`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `TraceMap`. F: ( alpha ) \mapsto \mathtt{TraceMap}(alpha).

##### 1.12-113 AddCoRankMorphism

`‣ AddCoRankMorphism`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoRankMorphism`. F: ( a ) \mapsto \mathtt{CoRankMorphism}(a).

##### 1.12-114 AddCoTraceMap

`‣ AddCoTraceMap`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoTraceMap`. F: ( alpha ) \mapsto \mathtt{CoTraceMap}(alpha).

##### 1.12-115 AddCoclosedCoevaluationForCoDual

`‣ AddCoclosedCoevaluationForCoDual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoclosedCoevaluationForCoDual`. F: ( a ) \mapsto \mathtt{CoclosedCoevaluationForCoDual}(a).

##### 1.12-116 AddCoclosedCoevaluationForCoDualWithGivenTensorProduct

`‣ AddCoclosedCoevaluationForCoDualWithGivenTensorProduct`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `CoclosedCoevaluationForCoDualWithGivenTensorProduct`. F: ( s, a, r ) \mapsto \mathtt{CoclosedCoevaluationForCoDualWithGivenTensorProduct}(s, a, r).

##### 1.12-117 AddInternalCoHomTensorProductCompatibilityMorphismInverse

`‣ AddInternalCoHomTensorProductCompatibilityMorphismInverse`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalCoHomTensorProductCompatibilityMorphismInverse`. F: ( list ) \mapsto \mathtt{InternalCoHomTensorProductCompatibilityMorphismInverse}(list).

##### 1.12-118 AddInternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects

`‣ AddInternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects`. F: ( source, list, range ) \mapsto \mathtt{InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects}(source, list, range).

##### 1.12-119 AddIsomorphismFromInternalCoHomToTensorProductWithCoDualObject

`‣ AddIsomorphismFromInternalCoHomToTensorProductWithCoDualObject`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromInternalCoHomToTensorProductWithCoDualObject`. F: ( a, b ) \mapsto \mathtt{IsomorphismFromInternalCoHomToTensorProductWithCoDualObject}(a, b).

##### 1.12-120 AddIsomorphismFromTensorProductWithCoDualObjectToInternalCoHom

`‣ AddIsomorphismFromTensorProductWithCoDualObjectToInternalCoHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom`. F: ( a, b ) \mapsto \mathtt{IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom}(a, b).

##### 1.12-121 AddMorphismFromTensorProductToInternalCoHom

`‣ AddMorphismFromTensorProductToInternalCoHom`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromTensorProductToInternalCoHom`. F: ( a, b ) \mapsto \mathtt{MorphismFromTensorProductToInternalCoHom}(a, b).

##### 1.12-122 AddMorphismFromTensorProductToInternalCoHomWithGivenObjects

`‣ AddMorphismFromTensorProductToInternalCoHomWithGivenObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismFromTensorProductToInternalCoHomWithGivenObjects`. F: ( s, a, b, r ) \mapsto \mathtt{MorphismFromTensorProductToInternalCoHomWithGivenObjects}(s, a, b, r).

##### 1.12-123 AddMorphismToCoBidual

`‣ AddMorphismToCoBidual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismToCoBidual`. F: ( a ) \mapsto \mathtt{MorphismToCoBidual}(a).

##### 1.12-124 AddMorphismToCoBidualWithGivenCoBidual

`‣ AddMorphismToCoBidualWithGivenCoBidual`( C, F ) | ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation `MorphismToCoBidualWithGivenCoBidual`. F: ( a, r ) \mapsto \mathtt{MorphismToCoBidualWithGivenCoBidual}(a, r).

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