Corollary 7.5.9.3. Let $\operatorname{\mathcal{C}}$ be a small filtered category. Then the inclusion map
preserves $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$-indexed colimits.
Proof. We first observe that the full subcategory $\operatorname{QCat}\subseteq \operatorname{Set_{\Delta }}$ is closed under filtered colimits (Remark 1.4.0.9), so the category $\operatorname{QCat}$ admits $\operatorname{\mathcal{C}}$-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 7.5.9.1. $\square$