Kerodon

Corollary 7.6.1.22. Let $\{ K_ i \} _{i \in I}$ be a collection of simplicial sets having coproduct $K = {\coprod }_{i \in I} K_ i$, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Suppose that $\operatorname{\mathcal{C}}$ admits $I$-indexed products and $K_ i$-indexed limits for each $i \in I$. Then $\operatorname{\mathcal{C}}$ admits $K$-indexed limits. Moreover, if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor which preserves $I$-indexed products and $K_ i$-indexed colimits for each $i \in I$, then $F$ also preserves $K$-indexed limits.