Kerodon

Remark 10.2.4.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which preserves finite products. Then the induced functor $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }})^{\operatorname{op}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }})^{\operatorname{op}}, \operatorname{\mathcal{D}})$ carries Čechnerves to Čechnerves. In particular, if $X$ is an object of $\operatorname{\mathcal{C}}$ which admits a Čechnerve $\operatorname{\check{C}}_{\bullet }(X)$, then the image $Y = F(X)$ also admits a Čechnerve, given by $\operatorname{\check{C}}_{\bullet }(Y) = F( \operatorname{\check{C}}_{\bullet }(X) )$.