GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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\begin{sequent}
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\begin{align*}
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  A:\Obj ~|~ \IsZero( A ) \vdash \IsTerminal( A )
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\end{align*}
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\end{sequent}
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\begin{sequent}
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\begin{align*}
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  A:\Obj ~|~ \IsZero( A ) \vdash \IsInitial( A )
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\end{align*}
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\end{sequent}
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\begin{sequent}
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\begin{align*}
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  A:\Obj ~|~ \IsZero( A ) \vdash \IsInjective( A )
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\end{align*}
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\end{sequent}
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\begin{sequent}
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\begin{align*}
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  A:\Obj ~|~ \IsZero( A ) \vdash \IsProjective( A )
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\end{align*}
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\end{sequent}
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\begin{sequent}
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\begin{align*}
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  \alpha:\Mor ~|~ \IsIsomorphism( \alpha ) \vdash \IsSplitMonomorphism( \alpha )
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\end{align*}
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\end{sequent}
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\begin{sequent}
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\begin{align*}
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  \alpha:\Mor ~|~ \IsIsomorphism( \alpha ) \vdash \IsSplitEpimorphism( \alpha )
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\end{align*}
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\end{sequent}
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\begin{sequent}
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\begin{align*}
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  \alpha:\Mor ~|~ \IsOne( \alpha ) \vdash \IsAutomorphism( \alpha )
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\end{align*}
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\end{sequent}
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\begin{sequent}
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\begin{align*}
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  \alpha:\Mor ~|~ \IsAutomorphism( \alpha ) \vdash \IsIsomorphism( \alpha )
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\end{align*}
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\end{sequent}
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\begin{sequent}
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\begin{align*}
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  \alpha:\Mor ~|~ \IsAutomorphism( \alpha ) \vdash \IsEndomorphism( \alpha )
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\end{align*}
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\end{sequent}
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\begin{sequent}
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\begin{align*}
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  \alpha:\Mor ~&|~  \IsEndomorphism( \alpha ), \IsIsomorphism( \alpha ) \\ 
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  &\vdash \IsAutomorphism( \alpha )
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\end{align*}
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\end{sequent}
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\begin{sequent}
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\begin{align*}
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  \alpha:\Mor ~|~ \IsSplitMonomorphism( \alpha ) \vdash \IsMonomorphism( \alpha )
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\end{align*}
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\end{sequent}
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\begin{sequent}
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\begin{align*}
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  \alpha:\Mor ~|~ \IsSplitEpimorphism( \beta ) \vdash \IsEpimorphism( \beta )
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\end{align*}
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\end{sequent}
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