Monoidal Categories

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.

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. 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. 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. 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. 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). 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). 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. 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. 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. 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. 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. 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. 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. 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. an object The arguments are two objects a, b. The output is the tensor product a \otimes b. 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. an object The argument is a category \mathbf{C}. The output is the tensor unit 1 of \mathbf{C}. 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.
Additive Monoidal Categories

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). 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. 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). 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. 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). 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. 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 . 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.
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.

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. 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. 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. 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.
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.

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.

an object The arguments are two objects a,b. The output is the internal hom object \mathrm{\underline{Hom}}(a,b). 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'). 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'). 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. 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. 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. 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. 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. 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. 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. 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. 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). 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). 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). 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). an object The argument is an object a. The output is its dual object a^{\vee}. 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}. 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}. 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. 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. 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}. 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}. 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'). 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'). 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}. 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}. 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). 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). 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). 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}. 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}. 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. 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. 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). 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). 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. 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.
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.

an object The arguments are two objects a,b. The output is the internal cohom object \mathrm{\underline{coHom}}(a,b). 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). 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). 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. 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. 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. 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. 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. 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. 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. 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. 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). 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). 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). 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). an object The argument is an object a. The output is its codual object a_{\vee}. 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}. 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}. 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. 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. 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. 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. 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'). 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'). 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}. 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}. 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. 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}. 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). 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}. 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}. 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. 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. 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). 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). 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. 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.
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.

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.

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.

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). 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. 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. 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. 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'). 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'). 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}. 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}. 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. 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. 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. 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.
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.

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. 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). 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). 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). 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'). 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'). 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. 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. 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. 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. 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}. 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}.
Convenience Methods

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. 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.
Add-methods

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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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). 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).