Remark 2.2.6.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isomorphism of $2$-categories, and let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be the inverse isomorphism. Then:
The functor $F$ is strictly unitary if and only if $G$ is strictly unitary. In this case, we say that $F$ is a strictly unitary isomorphism.
The functor $F$ is strict if and only if $G$ is strict. In this case, we say that $F$ is a strict isomorphism.
We say that $2$-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are strictly isomorphic if there is a strict isomorphism from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$.