Corollary 9.1.10.9 (0636)

Corollary 9.1.10.9. Let $\kappa $ be a small regular cardinal, let $\mathbb {K}$ be a collection of $\kappa $-small simplicial sets, and let $\operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{cocont}}$ be the subcategory of $\operatorname{\mathcal{QC}}$ whose objects are $\mathbb {K}$-cocomplete $\infty $-categories and whose morphisms are $\mathbb {K}$-cocontinuous functors (see Notation 8.7.3.7). Then:

$(1)$: The $\infty $-category $\operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{cocont}}$ admits small $\kappa $-filtered colimits.
$(2)$: The inclusion functor $\operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{cocont}} \hookrightarrow \operatorname{\mathcal{QC}}$ is $\kappa $-finitary: that is, it preserves small $\kappa $-filtered colimits.

Proof. Suppose we are given a diagram $\mathscr {F}: \operatorname{\mathcal{J}}\rightarrow \operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{cocont}}$, where $\operatorname{\mathcal{J}}$ is a small $\kappa $-filtered $\infty $-category. It follows from Corollary 7.4.5.3 that $\mathscr {F}$ can be extended to a colimit diagram $\overline{\mathscr {F}}: \operatorname{\mathcal{J}}^{\triangleright } \rightarrow \operatorname{\mathcal{QC}}$. Using Propositions 9.1.10.5 and 9.1.10.8, we see that any such extension factors through the subcategory $\operatorname{\mathcal{QC}}^{ \mathbb {K}-\mathrm{cocont}} \subseteq \operatorname{\mathcal{QC}}$, where it is also a colimit diagram. $\square$