Example 6.1.0.3 (02CD)

Example 6.1.0.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors between categories, and let $\{ \rho _{C,D} \} _{C \in \operatorname{\mathcal{C}}, D \in \operatorname{\mathcal{D}}}$ be a $\operatorname{Hom}$-adjunction between $F$ and $G$ (in the sense of Definition 6.1.0.1). Let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ be the natural transformations given by the formulae

\[ \eta _ C = \rho _{C,F(C)}( \operatorname{id}_{F(C)} ) \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, (G \circ F)(C) ) \]

\[ \epsilon _ D = \rho _{G(D),D}^{-1}( \operatorname{id}_{G(D)} ) \in \operatorname{Hom}_{\operatorname{\mathcal{D}}}( (F \circ G)(D), D). \]

Then the pair $(\eta , \epsilon )$ is an adjunction between $F$ and $G$ (in the sense of Definition 6.1.0.2). Condition $(Z1)$ follows from the observation that for each object $C \in \operatorname{\mathcal{C}}$, we have

\begin{eqnarray*} \operatorname{id}_{F(C)} & = & \rho _{C, F(C)}^{-1}(\rho _{C,F(C)}( \operatorname{id}_{F(C)} )) \\ & = & \rho _{C, F(C)}^{-1}( \eta _ C ) \\ & = & \rho _{C, F(C)}^{-1}( \operatorname{id}_{(G \circ F)(C)} \circ \eta _ C) \\ & = & \rho _{(G \circ F)(C), F(C)}^{-1}( \operatorname{id}_{(G \circ F)(C)}) \circ F(\eta _ C) \\ & = & \epsilon _{ F(C)} \circ F(\eta _ C). \end{eqnarray*}

The verification of $(Z2)$ is similar.