Notation 10.2.4.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$ which admits a Čechnerve $X_{\bullet }$. It follows from Corollary 7.3.6.13 that, for every simplicial object $Y_{\bullet }$ of $\operatorname{\mathcal{C}}$, the restriction map
\[ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{\mathcal{C}}) }( Y_{\bullet }, X_{\bullet } ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y_0, X_0) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y_0, X) \]
is a homotopy equivalence. In particular, the simplicial object $X_{\bullet }$ is unique up to isomorphism and depends functorially on $X$. To emphasize this dependence, we will denote $X_{\bullet }$ by $\operatorname{\check{C}}_{\bullet }(X)$ and refer to it as the Čechnerve of the object $X$.