Remark 9.3.1.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. Then an object $X \in \operatorname{\mathcal{C}}$ is $n$-truncated if and only if the right fibration $\operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}$ is essentially $n$-categorical. This follows from the criterion of Corollary 5.1.5.18 (together with Remark 9.3.1.2).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$