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

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

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

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

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

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

\begin{sequent}
\begin{align*}
  \alpha:\Mor ~|~ \IsOne( \alpha ) \vdash \IsAutomorphism( \alpha )
\end{align*}
\end{sequent}

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

\begin{sequent}
\begin{align*}
  \alpha:\Mor ~|~ \IsAutomorphism( \alpha ) \vdash \IsEndomorphism( \alpha )
\end{align*}
\end{sequent}

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

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

\begin{sequent}
\begin{align*}
  \alpha:\Mor ~|~ \IsSplitEpimorphism( \beta ) \vdash \IsEpimorphism( \beta )
\end{align*}
\end{sequent}