Lemma 9.2.4.16. Let $\kappa $ and $\lambda $ be regular cardinals, where $\lambda $ is uncountable. Then, for every $\infty $-category $\operatorname{\mathcal{C}}$, the full subcategory $\operatorname{Fun}^{\kappa -\flat }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } ) \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$ is closed under $\lambda $-small $\kappa $-filtered colimits.
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Proof. Enlarging $\lambda $ if necessary, we may assume that $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small. Let $T: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda }) \rightarrow \operatorname{\mathcal{QC}}_{< \lambda }$ be as in Lemma 9.2.4.15. Then a functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ belongs to $\operatorname{Fun}^{\kappa -\flat }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$ if and only if the $\infty $-category $T( \mathscr {F} )$ is $\kappa $-filtered (Remark 9.2.4.13). Since $T$ is $\lambda $-cocontinuous, the collection of objects which satisfy this condition is closed under $\lambda $-small, $\kappa $-filtered colimits (Corollary 9.1.5.13). $\square$