Let $\mathcal{C}$, $\mathcal{D}$, and $\mathcal{E}$ be categories.
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Functionality. The assignment $\webleft (\beta ,\alpha \webright )\mapsto \beta \circ \alpha $ defines a function
\[ \circ _{F,G,H}\colon \text{Nat}\webleft (G,H\webright )\times \text{Nat}\webleft (F,G\webright )\to \text{Nat}\webleft (F,H\webright ). \]
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Associativity. Let $F,G,H,K\colon \mathcal{C}\overset {\rightrightarrows }{\rightrightarrows }\mathcal{D}$ be functors. The diagram commutes, i.e. given natural transformations
\[ F\mathbin {\overset {\alpha }{\Longrightarrow }}G\mathbin {\overset {\beta }{\Longrightarrow }}H\mathbin {\overset {\gamma }{\Longrightarrow }}K, \]we have
\[ \webleft (\gamma \circ \beta \webright )\circ \alpha =\gamma \circ \webleft (\beta \circ \alpha \webright ). \] -
Unitality. Let $F,G\colon \mathcal{C}\rightrightarrows \mathcal{D}$ be functors.
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Left Unitality. The diagram
commutes, i.e. given a natural transformation $\alpha \colon F\Longrightarrow G$, we have
\[ \text{id}_{G}\circ \alpha =\alpha . \] -
Right Unitality. The diagram
commutes, i.e. given a natural transformation $\alpha \colon F\Longrightarrow G$, we have
\[ \alpha \circ \text{id}_{F}=\alpha . \]
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Left Unitality. The diagram
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Middle Four Exchange. Let $F_{1},F_{2},F_{3}\colon \mathcal{C}\to \mathcal{D}$ and $G_{1},G_{2},G_{3}\colon \mathcal{D}\to \mathcal{E}$ be functors. The diagram commutes, i.e. given a diagram
in $\mathsf{Cats}_{\mathsf{2}}$, we have
\[ \webleft (\beta '\mathbin {\star }\alpha '\webright )\circ \webleft (\beta \mathbin {\star }\alpha \webright )=\webleft (\beta '\circ \beta \webright )\mathbin {\star }\webleft (\alpha '\circ \alpha \webright ). \]
