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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%  \alpha:\Mor~|~ &() \\
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%  \vdash &\IsIsomorphism\big( \KernelLift( \\
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%  &\CokernelProjection( \alpha ), \CokernelColift( \KernelEmbedding( \alpha ), \alpha ))\big)
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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~|~ &() \\
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%  \vdash &\IsIsomorphism\big( \CokernelColift( \\
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%  &\KernelEmbedding( \alpha ), \KernelLift( \CokernelProjection( \alpha ), \alpha ))\big)
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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~|~ & \IsEpimorphism\big( \alpha \big) \\
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  \vdash &\IsIsomorphism\big( \ImageEmbedding( \alpha ) \big)
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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\big( \ImageEmbedding( \alpha ) \big) \\
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  \vdash &\IsEpimorphism\big( \alpha \big)
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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~|~ & \IsEpimorphism\big( \alpha \big) \\
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  \vdash &\IsIsomorphism\big( \AstrictionToCoimage( \alpha ) \big)
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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\big( \AstrictionToCoimage( \alpha ) \big) \\
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  \vdash &\IsEpimorphism\big( \alpha \big)
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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~|~ & \IsMonomorphism\big( \alpha \big) \\
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  \vdash &\IsIsomorphism\big( \CoimageProjection( \alpha ) \big)
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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\big( \CoimageProjection( \alpha ) \big) \\
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  \vdash &\IsMonomorphism\big( \alpha \big)
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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~|~ & \IsMonomorphism\big( \alpha \big) \\
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  \vdash &\IsIsomorphism\big( \CoastrictionToImage( \alpha ) \big)
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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\big( \CoastrictionToImage( \alpha ) \big) \\
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  \vdash &\IsMonomorphism\big( \alpha \big)
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\end{align*}
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\end{sequent}
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