Remark 2.2.8.16 (Functoriality). Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $2$-categories. Then there is a unique functor of ordinary categories $\operatorname{hPith}(U): \operatorname{hPith}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{hPith}(\operatorname{\mathcal{D}})$ with the following properties:
For each object $X \in \operatorname{\mathcal{C}}$, the functor $\operatorname{hPith}(U)$ carries $X$ to the object $U(X) \in \operatorname{\mathcal{D}}$.
For each $1$-morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$, the functor $\operatorname{hPith}(U)$ carries the isomorphism class $[f]$ to the isomorphism class of the $1$-morphism $U(f): U(X) \rightarrow U(Y)$.
Beware that the analogous assertion does not hold if $U$ is only assumed to be a lax functor of $2$-categories.