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

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1
\begin{sequent}

2
\begin{align*}

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  A:\Obj ~|~ () \vdash \IsIdempotent( \IdentityMorphism( 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 ~|~ () \vdash \IsOne\big( \IdentityMorphism( A ) \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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  A:\Obj ~|~ () \vdash \IsIdenticalToIdentityMorphism\big( \IdentityMorphism( A ) \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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   ~|~ () \vdash \IsZero\big( \ZeroObject() ) \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, \beta:\Mor  ~&|~ \IsMonomorphism( \beta ) \\

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  &\vdash \IsMonomorphism\big( \ProjectionInFactorOfFiberProduct( [\alpha, \beta], 1 ) \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, \beta:\Mor ~&|~ \IsMonomorphism( \alpha ) \\

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  &\vdash \IsMonomorphism\big( \ProjectionInFactorOfFiberProduct( [\alpha, \beta], 2 ) \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, \beta:\Mor ~&|~ \IsEpimorphism( \alpha ) \\

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  &\vdash \IsEpimorphism\big( \InjectionOfCofactorOfPushout( [\alpha, \beta], 2 ) \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, \beta:\Mor  ~&|~ \IsEpimorphism( \beta ) \\

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  &\vdash \IsEpimorphism\big( \InjectionOfCofactorOfPushout( [\alpha, \beta], 1 ) \big)

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\end{align*}

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\end{sequent}

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53
\begin{sequent}

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\begin{align*}

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  \alpha:\Mor ~&|~ () \\

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  &\vdash \IsMonomorphism\big( \KernelEmbedding( \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 \IsEpimorphism\big( \CokernelProjection( \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, \beta:\Mor  ~&|~ () \\

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  &\vdash \IsMonomorphism\big( \Equalizer( \alpha, \beta ) \big)

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\end{align*}

72
\end{sequent}

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\begin{sequent}

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\begin{align*}

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  \alpha:\Mor, \beta:\Mor  ~&|~ () \\

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  &\vdash \IsEpimorphism\big( \Coequalizer( \alpha, \beta ) \big)

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\end{align*}

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\end{sequent}

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\begin{sequent}\label{sequent:no_proper_context_7}

82
\begin{align*}

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  \alpha:\Mor ~&|~ \IsTerminal\big( \Source( \alpha ) \big)\\

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  &\vdash \IsSplitMonomorphism( \alpha )

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\end{align*}

86
\end{sequent}

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\begin{sequent}\label{sequent:no_proper_context_8}

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\begin{align*}

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  \alpha:\Mor ~&|~ \IsInitial\big( \Range( \alpha ) \big)\\

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  &\vdash \IsSplitEpimorphism( \alpha )

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\end{align*}

93
\end{sequent}

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95
\begin{sequent}

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\begin{align*}

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  L: \ListObj ~|~ \big(\forall x \in L: \IsTerminal(x)\big) \vdash \IsTerminal\big( \DirectProduct( L ) \big)

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\end{align*}

99
\end{sequent}

100


101
\begin{sequent}

102
\begin{align*}

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  L: \ListObj ~|~ \big(\forall x \in L: \IsInitial(x)\big) \vdash \IsInitial\big( \Coproduct( L ) \big)

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\end{align*}

105
\end{sequent}

106


107
\begin{sequent}

108
\begin{align*}

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 \alpha:\Mor, \beta:\Mor ~&|~ \IsMonomorphism( \alpha ), \IsMonomorphism( \beta )  \\

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 &\vdash \IsMonomorphism\big( \PreCompose( \alpha, \beta ) \big)

111
\end{align*}

112
\end{sequent}

113


114
\begin{sequent}

115
\begin{align*}

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 \alpha:\Mor, \beta:\Mor ~&|~ \IsEpimorphism( \alpha ), \IsEpimorphism( \beta )  \\

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 &\vdash \IsEpimorphism\big( \PreCompose( \alpha, \beta ) \big)

118
\end{align*}

119
\end{sequent}

120


121
\begin{sequent}

122
\begin{align*}

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 \alpha:\Mor ~&|~ \IsIsomorphism( \alpha ) &\vdash \IsIsomorphism( \InverseImmutable( \alpha ) )

124
\end{align*}

125
\end{sequent}

126


127
\begin{sequent}

128
\begin{align*}

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  \alpha:\Mor~|~ & () \\

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  \vdash &\IsMonomorphism\big( \ImageEmbedding( \alpha ) \big)

131
\end{align*}

132
\end{sequent}

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