Kerodon

Example 6.2.1.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories, and let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a homotopy inverse of $F$. Then $G$ is also a right adjoint of $F$. More precisely, any isomorphism $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ in the functor $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$ is the unit of an adjunction between $F$ and $G$ (Proposition 6.1.4.1). Similarly, $G$ is a left adjoint of $F$.