The category of (small) categories and functors is the category $\mathsf{Cats}$ where
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Objects. The objects of $\mathsf{Cats}$ are small categories.
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Morphisms. For each $\mathcal{C},\mathcal{D}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$, we have
\[ \operatorname {\mathrm{Hom}}_{\mathsf{Cats}}\webleft (\mathcal{C},\mathcal{D}\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{Obj}}\webleft (\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )\webright ). \] -
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Identities. For each $\mathcal{C}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$, the unit map
\[ \mathbb {1}^{\mathsf{Cats}}_{\mathcal{C}} \colon \mathrm{pt}\to \operatorname {\mathrm{Hom}}_{\mathsf{Cats}}\webleft (\mathcal{C},\mathcal{C}\webright ) \]of $\mathsf{Cats}$ at $\mathcal{C}$ is defined by
\[ \operatorname {\mathrm{id}}^{\mathsf{Cats}}_{\mathcal{C}} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{id}}_{\mathcal{C}}, \]where $\operatorname {\mathrm{id}}_{\mathcal{C}}\colon \mathcal{C}\to \mathcal{C}$ is the identity functor of $\mathcal{C}$ of Example 11.5.1.1.4.
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Composition. For each $\mathcal{C},\mathcal{D},\mathcal{E}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$, the composition map
\[ \circ ^{\mathsf{Cats}}_{\mathcal{C},\mathcal{D},\mathcal{E}} \colon \operatorname {\mathrm{Hom}}_{\mathsf{Cats}}\webleft (\mathcal{D},\mathcal{E}\webright ) \times \operatorname {\mathrm{Hom}}_{\mathsf{Cats}}\webleft (\mathcal{C},\mathcal{D}\webright ) \to \operatorname {\mathrm{Hom}}_{\mathsf{Cats}}\webleft (\mathcal{C},\mathcal{E}\webright ) \]of $\mathsf{Cats}$ at $\webleft (\mathcal{C},\mathcal{D},\mathcal{E}\webright )$ is given by
\[ G\mathbin {{\circ }^{\mathsf{Cats}}_{\mathcal{C},\mathcal{D},\mathcal{E}}}F \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}G\circ F, \]where $G\circ F\colon \mathcal{C}\to \mathcal{E}$ is the composition of $F$ and $G$ of Definition 11.5.1.1.5.
