Remark 10.2.5.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which preserves fiber products. Then the induced functor of augmented simplicial objects
\[ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}}), \operatorname{\mathcal{D}}) \]
carries Čechnerves to Čechnerves (see Remark 10.2.4.7). In particular, if $u: X \rightarrow Y$ is a morphism of $\operatorname{\mathcal{C}}$ which admits a Čechnerve $\operatorname{\check{C}}_{\bullet }(X/Y)$, then the morphism $F(u): F(X) \rightarrow F(Y)$ admits a Čechnerve in the $\infty $-category $\operatorname{\mathcal{D}}$, given by $F( \operatorname{\check{C}}_{\bullet }(X/Y) )$.