Proposition 9.1.10.1. Let $\mathscr {F}: \operatorname{\mathcal{J}}\rightarrow \operatorname{\mathcal{QC}}$ be a filtered diagram of $\infty $-categories. Assume that, for every object $J \in \operatorname{\mathcal{J}}$, the $\infty $-category $\mathscr {F}(J)$ is locally $n$-truncated, for some fixed integer $n$. Then the colimit $\varinjlim (\mathscr {F})$ is also locally $n$-truncated.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Using Theorem 9.1.8.1, we can choose a directed partially ordered set $(A, \leq )$ and a right cofinal functor $\operatorname{N}_{\bullet }(A) \rightarrow \operatorname{\mathcal{J}}$. Using Corollary 7.2.2.11, we can replace $\operatorname{\mathcal{J}}$ by $\operatorname{N}_{\bullet }(A)$ and thereby reduce to the case where $\operatorname{\mathcal{J}}$ is the nerve of a (directed) partially ordered set. Using Corollary 5.6.5.18, we can further reduce to the case where $\mathscr {F}$ is obtained from a strictly commutative diagram
\[ \mathscr {F}_0: (A, \leq ) \rightarrow \operatorname{QCat}\subset \operatorname{Set_{\Delta }}\quad \quad (\alpha \in A) \mapsto \operatorname{\mathcal{E}}_{\alpha }. \]
It follows from Corollary 9.1.6.3 that we can identify the colimit $\varinjlim (\mathscr {F})$ (formed in the $\infty $-category $\operatorname{\mathcal{QC}}$) with the colimit $\varinjlim (\mathscr {F}_0)$ (formed in the ordinary category of simplicial sets). The desired result now follows from Remark 4.8.2.4. $\square$