The category of functors from $\mathcal{C}$ to $\mathcal{D}$1 is the category $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$2 where
- Objects. The objects of $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$ are functors from $\mathcal{C}$ to $\mathcal{D}$.
- Morphisms. For each $F,G\in \text{Obj}\webleft (\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )\webright )$, we have
\[ \textup{Hom}_{\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )}\webleft (F,G\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{Nat}\webleft (F,G\webright ). \]
- Identities. For each $F\in \text{Obj}\webleft (\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )\webright )$, the unit map
\[ \mathbb {1}^{\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )}_{F} \colon \text{pt}\to \text{Nat}\webleft (F,F\webright ) \]
of $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$ at $F$ is given by
\[ \text{id}^{\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )}_{F} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{id}_{F}, \]where $\text{id}_{F}\colon F\Longrightarrow F$ is the identity natural transformation of $F$ of Example 9.9.3.1.1.
- Composition. For each $F,G,H\in \text{Obj}\webleft (\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )\webright )$, the composition map
\[ \circ ^{\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )}_{F,G,H} \colon \text{Nat}\webleft (G,H\webright ) \times \text{Nat}\webleft (F,G\webright ) \to \text{Nat}\webleft (F,H\webright ) \]
of $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$ at $\webleft (F,G,H\webright )$ is given by
\[ \beta \mathbin {{\circ }^{\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )}_{F,G,H}}\alpha \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\beta \circ \alpha , \]where $\beta \circ \alpha $ is the vertical composition of $\alpha $ and $\beta $ of Item 1 of Proposition 9.9.4.1.2.
1Further Terminology: Also called the functor category $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$.
2Further Notation: Also written $\mathcal{D}^{\mathcal{C}}$ and $\webleft [\mathcal{C},\mathcal{D}\webright ]$.
