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 $\mathcal{C},\mathcal{D}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$, we have
\[ \operatorname {\mathrm{Hom}}_{\mathsf{Cats}}(\mathcal{C},\mathcal{D}) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{Obj}}(\mathsf{Fun}(\mathcal{C},\mathcal{D})). \] -
•
Identities. For each $\mathcal{C}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$, the unit map
\[ \mathbb {1}^{\mathsf{Cats}}_{\mathcal{C}} \colon \mathrm{pt}\to \operatorname {\mathrm{Hom}}_{\mathsf{Cats}}(\mathcal{C},\mathcal{C}) \]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.
-
•
Composition. For each $\mathcal{C},\mathcal{D},\mathcal{E}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$, the composition map
\[ \circ ^{\mathsf{Cats}}_{\mathcal{C},\mathcal{D},\mathcal{E}} \colon \operatorname {\mathrm{Hom}}_{\mathsf{Cats}}(\mathcal{D},\mathcal{E}) \times \operatorname {\mathrm{Hom}}_{\mathsf{Cats}}(\mathcal{C},\mathcal{D}) \to \operatorname {\mathrm{Hom}}_{\mathsf{Cats}}(\mathcal{C},\mathcal{E}) \]of $\mathsf{Cats}$ at $(\mathcal{C},\mathcal{D},\mathcal{E})$ 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.
