Kerodon

Variant 10.3.2.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which admits a Čechnerve $\operatorname{\check{C}}(X/Y)_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}}_{+} ) \rightarrow \operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}^{0}_{/Y}$ denote the sieve generated by $f$. Then $\operatorname{\check{C}}(X/Y)_{\bullet }$ determines a right cofinal functor $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}^{0}_{/Y}$.