Corollary 9.5.1.12. Let $\kappa \leq \lambda $ be regular cardinals, where $\lambda $ is uncountable. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-compactly generated and $\lambda $-cocomplete, and suppose that the full subcategory $\operatorname{\mathcal{C}}_{< \kappa } \subseteq \operatorname{\mathcal{C}}$ spanned by the $(\kappa ,\lambda )$-compact objects is essentially $\lambda $-small. Let $K$ be a simplicial set. If the $\infty $-category $\operatorname{\mathcal{S}}_{< \lambda }$ admits $K$-indexed limits, then $\operatorname{\mathcal{C}}$ admits $K$-indexed limits.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Fix a regular cardinal $\mu $ such that $\operatorname{\mathcal{C}}$ is locally $\mu $-small and $\operatorname{\mathcal{S}}_{< \mu }$ admits $K$-indexed limits. As in the proof of Theorem 9.5.1.1, we can identify $\operatorname{\mathcal{C}}$ with the $\infty $-category $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } )$ spanned by the representable functors $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \mu }$. Since $\operatorname{\mathcal{S}}_{< \mu }$ admits $K$-indexed limits, the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } )$ has the same property (Proposition 7.1.8.2). It will therefore suffice to show that the full subcategory $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } )$ is closed under the formation of $K$-indexed limits. This follows from representability criterion of Proposition 9.5.1.11 (together with Remark 7.6.6.23). $\square$