Proposition 9.3.1.16 (Limits of Truncated Objects). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. Then the collection of $n$-truncated objects of $\operatorname{\mathcal{C}}$ is closed under limits. That is, if $F: A^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a limit diagram in $\operatorname{\mathcal{C}}$ having the property that $F(a)$ is $n$-truncated for each vertex $a \in A$, then $F$ carries the cone point of $A^{\triangleleft }$ to an $n$-truncated object of $\operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Fix an object $C \in \operatorname{\mathcal{C}}$ and let $h^{C}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ denote the functor corepresented by $C$. We wish to show that the composition
\[ A^{\triangleleft } \xrightarrow {F} \operatorname{\mathcal{C}}\xrightarrow {h^{C}} \operatorname{\mathcal{S}} \]
carries the cone point of $A^{\triangleleft }$ to an $n$-truncated Kan complex. This follows from Remark 7.4.1.5, since $h^{C} \circ F$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{S}}$ (Corollary 7.4.1.19). $\square$