Corollary 6.1.4.4. 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:
- $(1)$
The $1$-morphism $f$ is an isomorphism in $\operatorname{\mathcal{C}}$.
- $(2)$
The $1$-morphism $g$ is an isomorphism in $\operatorname{\mathcal{C}}$.
- $(3)$
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.