Kerodon

Remark 5.4.7.4 (Functoriality). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $(\infty ,2)$-categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Then $F$ is a functor (Definition 5.4.7.1) if and only it carries $\operatorname{Pith}(\operatorname{\mathcal{C}})$ into $\operatorname{Pith}(\operatorname{\mathcal{D}})$. If this condition is satisfied, then $\operatorname{Pith}(F) = F|_{ \operatorname{Pith}(\operatorname{\mathcal{C}}) }$ can be regarded as a functor from the $\infty $-category $\operatorname{Pith}(\operatorname{\mathcal{C}})$ to the $\infty $-category $\operatorname{Pith}(\operatorname{\mathcal{D}})$.