Remark 9.1.9.9. Let $\kappa \leq \lambda $ be regular cardinals, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits $\lambda $-small $\kappa $-filtered colimits. Let $\mu $ be an uncountable cardinal having cofinality $\geq \lambda $ and exponential cofinality $\geq \kappa $, so that the $\infty $-category $\operatorname{\mathcal{S}}_{< \mu }$ admits $\lambda $-small colimits and $\kappa $-small limits (see Corollary 7.4.3.8 and Variant 7.4.1.15). Then the $\infty $-category $\operatorname{Fun}^{(\kappa ,\lambda )-\operatorname{fin}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \mu } )$ is closed under the formation of $\kappa $-small limits in $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \mu } )$. See Proposition 9.1.5.8 (and Proposition 9.1.4.1). In particular:
If $\kappa $ is small and $\operatorname{\mathcal{C}}$ admits small $\kappa $-filtered colimits, then the collection of $\kappa $-finitary functors $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is closed under $\kappa $-small limits.
If $\operatorname{\mathcal{C}}$ admits small filtered colimits, then the collection of finitary functors $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is closed under finite limits.