Lemma 10.2.2.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$ which admits a Čechnerve $\operatorname{\check{C}}(X)_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ (see Definition 10.1.4.3). Let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be the sieve generated by $X$ (Example 10.2.1.19). Then the functor $\operatorname{\check{C}}(X)_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}^{0}$ is right cofinal.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $C$ be an object of $\operatorname{\mathcal{C}}$ and let $h^{C}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor corepresented by $C$. Since $h^{C}$ preserves finite products (Proposition 7.4.5.16), the composition
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \xrightarrow { \operatorname{\check{C}}(X)_{\bullet } } \operatorname{\mathcal{C}}\xrightarrow { h^{C} } \operatorname{\mathcal{S}} \]
is a simplicial object of $\operatorname{\mathcal{S}}$ which can be identified with the Čechnerve of the Kan complex $h^{C}( \operatorname{\check{C}}(X)_{0} ) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, X )$ (Remark 10.1.4.7). If $C$ belongs to the sieve $\operatorname{\mathcal{C}}^{0}$, then the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)$ is nonempty. Applying Corollary 10.1.6.15, we conclude that the geometric realization $| h^{C}( \operatorname{\check{C}}(X)_{\bullet } ) |$ is contractible. The desired result now follows by allowing the object $C$ to vary and applying the criterion of Proposition 7.4.5.11. $\square$