Corollary 9.3.1.22. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$ which is $n$-truncated for some integer $n$. If $K$ is an $(n+1)$-connective simplicial set, then a morphism $K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}_{/X}$ is a limit diagram if and only if its image in $\operatorname{\mathcal{C}}$ is a limit diagram. In particular, the forgetful functor $\operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}$ preserves $K$-indexed limits.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$