\begin{sequent}
\begin{align*}
   a: \Obj, b: \Obj ~|~ () \vdash \IsZeroForMorphisms \big( \ZeroMorphism( a, b ) \big)
\end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
   a: \Obj, b: \Obj ~|~ () \vdash \IsEqualToZeroMorphism \big( \ZeroMorphism( a, b ) \big)
\end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
   a: \Obj ~|~ () \vdash \IsZeroForMorphisms \big( \UniversalMorphismIntoZeroObject( a ) \big)
\end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
   a: \Obj ~|~ () \vdash \IsZeroForMorphisms \big( \UniversalMorphismFromZeroObject( a ) \big)
\end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
   \alpha: \Mor ~|~ \IsZeroForObjects(\Source(\alpha)) \vdash \IsZeroForMorphisms(\alpha)
\end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
   \alpha: \Mor ~|~ \IsZeroForObjects(\Range(\alpha)) \vdash \IsZeroForMorphisms(\alpha)
\end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
   \alpha: \Mor ~|~ \IsZeroForMorphisms(\alpha), \IsMonomorphism(\alpha) \vdash \IsZeroForObjects(\Source(\alpha))
\end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
   \alpha: \Mor ~|~ \IsZeroForMorphisms(\alpha), \IsEpimorphism(\alpha) \vdash \IsZeroForObjects(\Range(\alpha))
\end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
   \alpha: \Mor, \beta: \Mor ~|~ \IsZeroForMorphisms(\alpha) \vdash \IsZeroForMorphisms(\PreCompose(\alpha,\beta))
\end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
   \alpha: \Mor, \beta: \Mor ~|~ \IsZeroForMorphisms(\beta) \vdash \IsZeroForMorphisms(\PreCompose(\alpha,\beta))
\end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
  \alpha: \Mor ~|~ \IsInitial\big( \KernelObject( \alpha ) \big) \vdash \IsMonomorphism( \alpha )
\end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
  \alpha: \Mor ~|~ \IsMonomorphism( \alpha ) \vdash \IsZeroForObjects( \KernelObject( \alpha ) )
\end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
  \alpha: \Mor ~|~ \IsZeroForMorphisms\big( \KernelEmbedding( \alpha ) \big) \vdash \IsMonomorphism( \alpha )
\end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
  \alpha: \Mor ~|~ \IsMonomorphism( \alpha ) \vdash \IsZeroForMorphisms( \KernelEmbedding( \alpha ) )
\end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
  \alpha: \Mor ~|~ \IsTerminal\big( \CokernelObject( \alpha ) \big) \vdash \IsEpimorphism( \alpha )
\end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
  \alpha: \Mor ~|~ \IsEpimorphism( \alpha ) \vdash \IsZeroForObjects\big( \CokernelObject( \alpha ) \big)
\end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
  \alpha: \Mor ~|~ \IsZeroForMorphisms\big( \CokernelProjection( \alpha ) \big) \vdash \IsEpimorphism( \alpha )
\end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
  \alpha: \Mor ~|~  \IsEpimorphism( \alpha ) \vdash \IsZeroForMorphisms\big( \CokernelProjection( \alpha ) \big)
\end{align*}
\end{sequent}
