Exercise 6.1.0.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors between categories. Show that every adjunction $(\eta , \epsilon )$ between $F$ and $G$ can be obtained by applying the construction of Example 6.1.0.3 to a unique $\operatorname{Hom}$-adjunction $\{ \rho _{C,D} \} _{C \in \operatorname{\mathcal{C}}, D \in \operatorname{\mathcal{D}}}$ between $F$ and $G$ (see Example 6.1.2.7).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$