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

Corollary Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $f: C \rightarrow D$ and $g: D \rightarrow C$ be $1$-morphisms of $\operatorname{\mathcal{C}}$, and let $(\eta , \epsilon )$ be an adjunction between $f$ and $g$. The following conditions are equivalent:


The $1$-morphism $f$ is an isomorphism in $\operatorname{\mathcal{C}}$.


The $1$-morphism $g$ is an isomorphism in $\operatorname{\mathcal{C}}$.


The $2$-morphisms $\eta $ and $\epsilon $ are isomorphisms in $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,C)$ and $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,D)$, respectively. In particular, $f$ and $g$ are homotopy inverse to one another.

Proof. The implication $(3) \Rightarrow (1)$ and $(3) \Rightarrow (2)$ are immediate from the definitions, and the reverse implications follow by applying Proposition to $\operatorname{\mathcal{C}}$ and the conjugate $2$-category $\operatorname{\mathcal{C}}^{\operatorname{c}}$. $\square$