Example 6.1.2.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors between categories, and let $(\eta , \epsilon )$ be an adjunction between $F$ and $G$. Suppose we are given objects $C \in \operatorname{\mathcal{C}}$ and $D \in \operatorname{\mathcal{D}}$, which we identify with functors $C: \{ \ast \} \rightarrow \operatorname{\mathcal{C}}$ and $D: \{ \ast \} \rightarrow \operatorname{\mathcal{D}}$, respectively. Applying Corollary 6.1.2.6 to the $2$-category $\mathbf{Cat}$, we obtain a bijection
This bijection depends functorially on $C$ and $D$ (Remark 6.1.2.4), and can therefore be regarded as a $\operatorname{Hom}$-adjunction between $F$ and $G$ in the sense of Definition 6.1.0.1. Note that, for every morphism $f: F(C) \rightarrow D$ in $\operatorname{\mathcal{C}}$, the image $\rho _{C,D}(f) \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, G(D) )$ is given explicitly by the composition $C \xrightarrow { \eta _ C } (G \circ F)(C) \xrightarrow { G(f)} G(D)$. In particular, the morphism $\eta _ C: C \rightarrow (G \circ F)(C)$ can be recovered by applying $\rho _{C,F(C)}$ to the identity morphism $\operatorname{id}_{ F(C) }$. Similarly, for each object $D \in \operatorname{\mathcal{D}}$, the morphism $\epsilon _ D: (F \circ G)(D) \rightarrow D$ can be recovered by applying $\rho _{G(D), D}^{-1}$ to the identity morphism $\operatorname{id}_{G(D)}$. In other words, the adjunction $(\eta , \epsilon )$ is obtained by applying the construction of Example 6.1.0.3 to the $\operatorname{Hom}$-adjunction $\{ \rho _{C,D} \} _{C \in \operatorname{\mathcal{C}}, D \in \operatorname{\mathcal{D}}}$.