The $2$-category of (small) categories, functors, and natural transformations is the $2$-category $\mathsf{Cats}_{\mathsf{2}}$ where
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Objects. The objects of $\mathsf{Cats}_{\mathsf{2}}$ are small categories.
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$\mathsf{Hom}$-Categories. For each $\mathcal{C},\mathcal{D}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats}_{\mathsf{2}})$, we have
\[ \mathsf{Hom}_{\mathsf{Cats}_{\mathsf{2}}}(\mathcal{C},\mathcal{D}) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathsf{Fun}(\mathcal{C},\mathcal{D}). \] -
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Identities. For each $\mathcal{C}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats}_{\mathsf{2}})$, the unit functor
\[ \mathbb {1}^{\mathsf{Cats}_{\mathsf{2}}}_{\mathcal{C}} \colon \mathsf{pt}\to \mathsf{Fun}(\mathcal{C},\mathcal{C}) \]of $\mathsf{Cats}_{\mathsf{2}}$ at $\mathcal{C}$ is the functor picking the identity functor $\operatorname {\mathrm{id}}_{\mathcal{C}}\colon \mathcal{C}\to \mathcal{C}$ of $\mathcal{C}$.
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Composition. For each $\mathcal{C},\mathcal{D},\mathcal{E}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats}_{\mathsf{2}})$, the composition bifunctor
\[ \circ ^{\mathsf{Cats}_{\mathsf{2}}}_{\mathcal{C},\mathcal{D},\mathcal{E}} \colon \mathsf{Hom}_{\mathsf{Cats}_{\mathsf{2}}}(\mathcal{D},\mathcal{E}) \times \mathsf{Hom}_{\mathsf{Cats}_{\mathsf{2}}}(\mathcal{C},\mathcal{D}) \to \mathsf{Hom}_{\mathsf{Cats}_{\mathsf{2}}}(\mathcal{C},\mathcal{E}) \]of $\mathsf{Cats}_{\mathsf{2}}$ at $(\mathcal{C},\mathcal{D},\mathcal{E})$ is the functor where
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Action on Objects. For each object $(G,F)\in \operatorname {\mathrm{Obj}}(\mathsf{Hom}_{\mathsf{Cats}_{\mathsf{2}}}(\mathcal{D},\mathcal{E})\times \mathsf{Hom}_{\mathsf{Cats}_{\mathsf{2}}}(\mathcal{C},\mathcal{D}))$, we have
\[ \circ ^{\mathsf{Cats}_{\mathsf{2}}}_{\mathcal{C},\mathcal{D},\mathcal{E}}(G,F) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}G\circ F. \] -
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Action on Morphisms. For each morphism $(\beta ,\alpha )\colon (K,H)\Longrightarrow (G,F)$ of $\mathsf{Hom}_{\mathsf{Cats}_{\mathsf{2}}}(\mathcal{D},\mathcal{E})\times \mathsf{Hom}_{\mathsf{Cats}_{\mathsf{2}}}(\mathcal{C},\mathcal{D})$, we have
\[ \circ ^{\mathsf{Cats}_{\mathsf{2}}}_{\mathcal{C},\mathcal{D},\mathcal{E}}(\beta ,\alpha ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\beta \mathbin {\star }\alpha , \]where $\beta \mathbin {\star }\alpha $ is the horizontal composition of $\alpha $ and $\beta $ of Definition 11.9.5.1.1.
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