\begin{sequent}
\begin{align*}
  A:\Obj ~|~ () \vdash \IsIdempotent( \IdentityMorphism( A ) )
\end{align*}
\end{sequent}

\begin{sequent}
 \begin{align*}
  A:\Obj ~|~ () \vdash \IsOne\big( \IdentityMorphism( A ) \big)
 \end{align*}
\end{sequent}

\begin{sequent}
 \begin{align*}
  A:\Obj ~|~ () \vdash \IsIdenticalToIdentityMorphism\big( \IdentityMorphism( A ) \big)
 \end{align*}
\end{sequent}

\begin{sequent}
 \begin{align*}
   ~|~ () \vdash \IsZeroForObjects\big( \ZeroObject() ) \big)
 \end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
  \alpha:\Mor, \beta:\Mor  ~&|~ \IsMonomorphism( \beta ) \\
  &\vdash \IsMonomorphism\big( \ProjectionInFactorOfFiberProduct( [\alpha, \beta], 1 ) \big)
\end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
  \alpha:\Mor, \beta:\Mor ~&|~ \IsMonomorphism( \alpha ) \\
  &\vdash \IsMonomorphism\big( \ProjectionInFactorOfFiberProduct( [\alpha, \beta], 2 ) \big)
\end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
  \alpha:\Mor, \beta:\Mor ~&|~ \IsEpimorphism( \alpha ) \\
  &\vdash \IsEpimorphism\big( \InjectionOfCofactorOfPushout( [\alpha, \beta], 2 ) \big)
\end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
  \alpha:\Mor, \beta:\Mor  ~&|~ \IsEpimorphism( \beta ) \\
  &\vdash \IsEpimorphism\big( \InjectionOfCofactorOfPushout( [\alpha, \beta], 1 ) \big)
\end{align*}
\end{sequent}

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

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

\begin{sequent}
\begin{align*}
  L:\ListMor ~&|~ () \\
  &\vdash \IsMonomorphism\big( \EmbeddingOfEqualizer( L ) \big)
\end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
  L:\ListMor ~&|~ () \\
  &\vdash \IsEpimorphism\big( \ProjectionOntoCoequalizer( L ) \big)
\end{align*}
\end{sequent}

\begin{sequent}\label{sequent:no_proper_context_7}
\begin{align*}
  \alpha:\Mor ~&|~ \IsTerminal\big( \Source( \alpha ) \big)\\
  &\vdash \IsSplitMonomorphism( \alpha )
\end{align*}
\end{sequent}

\begin{sequent}\label{sequent:no_proper_context_8}
\begin{align*}
  \alpha:\Mor ~&|~ \IsInitial\big( \Range( \alpha ) \big)\\
  &\vdash \IsSplitEpimorphism( \alpha )
\end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
  L: \ListObj ~|~ \big(\forall x \in L: \IsTerminal(x)\big) \vdash \IsTerminal\big( \DirectProduct( L ) \big)
\end{align*}
\end{sequent}

\begin{sequent}
\begin{align*}
  L: \ListObj ~|~ \big(\forall x \in L: \IsInitial(x)\big) \vdash \IsInitial\big( \Coproduct( L ) \big)
\end{align*}
\end{sequent}

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

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

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

\begin{sequent}
\begin{align*}
 \alpha:\Mor ~&|~ \IsIsomorphism( \alpha ) &\vdash \IsIsomorphism( \InverseImmutable( \alpha ) )
\end{align*}
\end{sequent}

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