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

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