$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary Suppose we are given a diagram of simplicial sets

\[ K(0) \rightarrow K(1) \rightarrow K(2) \rightarrow K(3) \rightarrow \cdots \]

having colimit $K$, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits sequential limits and $K(n)$-indexed limits, for each $n \geq 0$. Then $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits. If $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories which preserves sequential limits and $K(n)$-indexed limits for each $n \geq 0$, then it also preserves $K$-indexed limits.