Proposition 10.2.4.5 (Existence). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. Then $X$ admits a Čechnerve $\operatorname{\check{C}}_{\bullet }(X)$ if and only if, for every nonempty finite set $J$, there exists a product of $J$ copies of $X$ in the $\infty $-category $\operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. For every integer $n \geq 0$, the category $\{ [0] \} \times _{\operatorname{{\bf \Delta }}} \operatorname{{\bf \Delta }}_{/ [n] }$ is isomorphic to the finite set $\{ 0, 1, \cdots , n \} $, regarded as a category having only identity morphisms. By virtue of Remark 10.2.4.2, the desired result is a special case of the existence criterion for Kan extensions (Corollary 7.3.5.8). $\square$