Kerodon

Remark 5.4.7.5. 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. If $F$ is a functor of $(\infty ,2)$-categories and $u: X \rightarrow Y$ is an isomorphism in the $(\infty ,2)$-category $\operatorname{\mathcal{C}}$, then $F(u): F(X) \rightarrow F(Y)$ is an isomorphism in the $(\infty ,2)$-category $\operatorname{\mathcal{D}}$ (see Definition 5.4.5.12). This follows by applying Remark 1.5.1.6 to the functor $\operatorname{Pith}(F): \operatorname{Pith}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Pith}(\operatorname{\mathcal{D}})$ of Remark 5.4.7.4. Beware that, if $F$ is not assumed to be a functor, then $F(u)$ need not be an isomorphism.