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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, b: \Obj ~|~ () \vdash \IsZero \big( \ZeroMorphism( a, b ) \big)

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\end{align*}

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\end{sequent}

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\begin{sequent}

8
\begin{align*}

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   a: \Obj, b: \Obj ~|~ () \vdash \IsIdenticalToZeroMorphism \big( \ZeroMorphism( a, b ) \big)

10
\end{align*}

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\end{sequent}

12


13
\begin{sequent}

14
\begin{align*}

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   a: \Obj ~|~ () \vdash \IsZero \big( \UniversalMorphismIntoZeroObject( a ) \big)

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\end{align*}

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\end{sequent}

18


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\begin{sequent}

20
\begin{align*}

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   a: \Obj ~|~ () \vdash \IsZero \big( \UniversalMorphismFromZeroObject( 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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   \alpha: \Mor ~|~ \IsZero(\Source(\alpha)) \vdash \IsZero(\alpha)

28
\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 ~|~ \IsZero(\Range(\alpha)) \vdash \IsZero(\alpha)

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\end{align*}

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\end{sequent}

36


37
\begin{sequent}

38
\begin{align*}

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   \alpha: \Mor ~|~ \IsZero(\alpha), \IsMonomorphism(\alpha) \vdash \IsZero(\Source(\alpha))

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\end{align*}

41
\end{sequent}

42


43
\begin{sequent}

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\begin{align*}

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   \alpha: \Mor ~|~ \IsZero(\alpha), \IsEpimorphism(\alpha) \vdash \IsZero(\Range(\alpha))

46
\end{align*}

47
\end{sequent}

48


49
\begin{sequent}

50
\begin{align*}

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   \alpha: \Mor, \beta: \Mor ~|~ \IsZero(\alpha) \vdash \IsZero(\PreCompose(\alpha,\beta))

52
\end{align*}

53
\end{sequent}

54


55
\begin{sequent}

56
\begin{align*}

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   \alpha: \Mor, \beta: \Mor ~|~ \IsZero(\beta) \vdash \IsZero(\PreCompose(\alpha,\beta))

58
\end{align*}

59
\end{sequent}

60


61
\begin{sequent}

62
\begin{align*}

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  \alpha: \Mor ~|~ \IsInitial\big( \KernelObject( \alpha ) \big) \vdash \IsMonomorphism( \alpha )

64
\end{align*}

65
\end{sequent}

66


67
\begin{sequent}

68
\begin{align*}

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  \alpha: \Mor ~|~ \IsMonomorphism( \alpha ) \vdash \IsZero( \KernelObject( \alpha ) )

70
\end{align*}

71
\end{sequent}

72


73
\begin{sequent}

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\begin{align*}

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  \alpha: \Mor ~|~ \IsZero\big( \KernelEmbedding( \alpha ) \big) \vdash \IsMonomorphism( \alpha )

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\end{align*}

77
\end{sequent}

78


79
\begin{sequent}

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\begin{align*}

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  \alpha: \Mor ~|~ \IsMonomorphism( \alpha ) \vdash \IsZero( \KernelEmbedding( \alpha ) )

82
\end{align*}

83
\end{sequent}

84


85
\begin{sequent}

86
\begin{align*}

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  \alpha: \Mor ~|~ \IsTerminal\big( \CokernelObject( \alpha ) \big) \vdash \IsEpimorphism( \alpha )

88
\end{align*}

89
\end{sequent}

90


91
\begin{sequent}

92
\begin{align*}

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  \alpha: \Mor ~|~ \IsEpimorphism( \alpha ) \vdash \IsZero\big( \CokernelObject( \alpha ) \big)

94
\end{align*}

95
\end{sequent}

96


97
\begin{sequent}

98
\begin{align*}

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  \alpha: \Mor ~|~ \IsZero\big( \CokernelProjection( \alpha ) \big) \vdash \IsEpimorphism( \alpha )

100
\end{align*}

101
\end{sequent}

102


103
\begin{sequent}

104
\begin{align*}

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  \alpha: \Mor ~|~  \IsEpimorphism( \alpha ) \vdash \IsZero\big( \CokernelProjection( \alpha ) \big)

106
\end{align*}

107
\end{sequent}

108


109