Proposition 9.1.4.3. Let $K$ and $L$ be small simplicial sets. Then $K$-indexed colimits commute with $L$-indexed limits in the $\infty $-category $\operatorname{\mathcal{S}}$ if and only if the following condition is satisfied:
- $(\ast )$
Let $\operatorname{Fun}'(K, \operatorname{\mathcal{S}})$ denote the full subcategory of $\operatorname{Fun}(K, \operatorname{\mathcal{S}})$ spanned by those diagrams $\mathscr {F}: K \rightarrow \operatorname{\mathcal{S}}$ for which the colimit $\varinjlim (\mathscr {F}) \in \operatorname{\mathcal{S}}$ is contractible. Then $\operatorname{Fun}'(K, \operatorname{\mathcal{S}})$ is closed under $L$-indexed limits.