Corollary 9.1.10.11. Let $\kappa < \lambda $ be regular cardinals, where $\lambda $ has exponential cofinality $\geq \kappa $. Let $\mathbb {K}$ be a collection of $\kappa $-small simplicial sets and let $T: \operatorname{\mathcal{QC}}_{< \lambda } \rightarrow \operatorname{\mathcal{QC}}_{< \lambda }$ be a $\mathbb {K}$-cocompletion functor (see Definition 8.7.3.3 and Proposition 8.7.3.5). Then $T$ is $(\kappa , \lambda )$-finitary: that is, it preserves $\lambda $-small $\kappa $-filtered colimits.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $\iota : \operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{cocont}}_{< \lambda } \hookrightarrow \operatorname{\mathcal{QC}}_{< \lambda }$ be the inclusion functor. By virtue of Proposition 8.7.3.9, the functor $T$ factors as a composition
\[ \operatorname{\mathcal{QC}}_{< \lambda } \xrightarrow {T_0} \operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{cocont}}_{< \lambda } \xrightarrow {\iota } \operatorname{\mathcal{QC}}_{< \lambda }, \]
where $T_0$ is left adjoint to $\iota $ and therefore preserves arbitrary colimits (Corollary 7.1.4.22). It will therefore suffice to show that the functor $\iota $ is $(\kappa , \lambda )$-finitary, which follows from Variant 9.1.10.10. $\square$