Corollary 9.1.7.3. The inclusion functor $\iota : \operatorname{N}_{\bullet }( \operatorname{QCat}) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}) = \operatorname{\mathcal{QC}}$ preserves small filtered colimits.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By virtue of Variant 9.1.6.8, it will suffice to show that, for every small filtered category $\operatorname{\mathcal{C}}$, the functor $\iota $ preserves $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$-indexed colimits. Fix 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 9.1.7.1. $\square$