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

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

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

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