Corollary 9.1.5.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\kappa $ be a singular cardinal. Then $\operatorname{\mathcal{C}}$ is $\kappa $-filtered if and only if it is $\kappa ^{+}$-filtered.
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Proof. Fix an uncountable regular cardinal $\lambda $ of exponential cofinality $\geq \kappa ^{+}$ such that $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small. It follows from Proposition 9.1.5.8, the colimit functor $\varinjlim : \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } ) \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ preserves $\kappa $-small limits, and we wish to show that it preserves $\kappa ^{+}$-small limits. By virtue of Corollary 7.6.6.11, it will suffice to show that if $I$ is set of cardinality $\kappa $, then $\varinjlim $ preserves $I$-indexed products. Our assumption that $\kappa $ is singular guarantees that we can decompose $I$ as a disjoint union $\coprod _{j \in J} I_{j}$, where $J$ is $\kappa $-small and each $I_{j}$ is $\kappa $-small. The desired result now follows from Corollary 7.6.1.22. $\square$