$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $f: C \rightarrow D$ be an isomorphism in $\operatorname{\mathcal{C}}$. Then any homotopy inverse to $f$ is both a left adjoint and a right adjoint of $f$.

Proof. Let $g: D \rightarrow C$ be a homotopy inverse to $f$, so that there exists an isomorphism $\eta : \operatorname{id}_{C} \xRightarrow {\sim } g \circ f$ in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,C)$. It follows from Proposition that $\eta $ is the unit of an adjunction, and therefore exhibits $g$ as a right adjoint to $f$. A similar argument shows that $g$ is left adjoint to $f$. $\square$