Remark 2.2.6.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $2$-categories. Then $F$ is an isomorphism (in the sense of Definition 2.2.6.1) if and only if it satisfies the following conditions:
The functor $F$ induces a bijection from the set of objects of $\operatorname{\mathcal{C}}$ to the set of objects of $\operatorname{\mathcal{D}}$.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the functor $F$ induces an isomorphism of categories $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$.