Definition 6.1.0.1 (Kan). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors. A $\operatorname{Hom}$-adjunction between $F$ and $G$ is a collection of bijections
which depend functorially on $C \in \operatorname{\mathcal{C}}$ and $D \in \operatorname{\mathcal{D}}$ (that is, the construction $(C,D) \mapsto \rho _{C,D}$ is an isomorphism in the functor category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{Set})$). In this case, we say that the construction $(C,D) \mapsto \rho _{C,D}$ exhibits $F$ as a left adjoint to $G$ and $G$ as a right adjoint to $F$.