The category of (small) categories and functors is the category $\mathsf{Cats}$ where
- Objects. The objects of $\mathsf{Cats}$ are small categories.
- Morphisms. For each $\mathcal{C},\mathcal{D}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, we have
\[ \textup{Hom}_{\mathsf{Cats}}\webleft (\mathcal{C},\mathcal{D}\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{Obj}\webleft (\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )\webright ). \]
- Identities. For each $\mathcal{C}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, the unit map
\[ \mathbb {1}^{\mathsf{Cats}}_{\mathcal{C}} \colon \text{pt}\to \textup{Hom}_{\mathsf{Cats}}\webleft (\mathcal{C},\mathcal{C}\webright ) \]
of $\mathsf{Cats}$ at $\mathcal{C}$ is defined by
\[ \text{id}^{\mathsf{Cats}}_{\mathcal{C}} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{id}_{\mathcal{C}}, \]where $\text{id}_{\mathcal{C}}\colon \mathcal{C}\to \mathcal{C}$ is the identity functor of $\mathcal{C}$ of Example 8.4.1.1.4.
- Composition. For each $\mathcal{C},\mathcal{D},\mathcal{E}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, the composition map
\[ \circ ^{\mathsf{Cats}}_{\mathcal{C},\mathcal{D},\mathcal{E}} \colon \textup{Hom}_{\mathsf{Cats}}\webleft (\mathcal{D},\mathcal{E}\webright ) \times \textup{Hom}_{\mathsf{Cats}}\webleft (\mathcal{C},\mathcal{D}\webright ) \to \textup{Hom}_{\mathsf{Cats}}\webleft (\mathcal{C},\mathcal{E}\webright ) \]
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 8.4.1.1.5.
