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

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

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

\begin{sequent}
\begin{align*}
   a: \Obj ~|~ () \vdash \IsZero \big( \MorphismFromZeroObject( a ) \big)
\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 \IsZero( \KernelObject( \alpha ) )
\end{align*}
\end{sequent}

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

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

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

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