Corollary 9.1.6.3. The inclusion functor $\iota : \operatorname{N}_{\bullet }( \operatorname{QCat}) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}) = \operatorname{\mathcal{QC}}$ preserves colimits indexed by small filtered categories.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $\operatorname{\mathcal{C}}$ be a small filtered category and suppose we are given a colimit diagram $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{QCat}$ in the ordinary category $\operatorname{QCat}$. We wish to show that the induced map $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \mathscr {F} ): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{QC}}$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}$. By virtue of Corollary 7.5.8.9, this is equivalent to the requirement that $\overline{\mathscr {F}}$ is a categorical colimit diagram, which follows from Proposition 9.1.6.1. $\square$