Corollary 7.1.6.20. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $K$ be a weakly contractible simplicial set. Then:
If $U$ is a right fibration and the $\infty $-category $\operatorname{\mathcal{D}}$ admits $K$-indexed limits, then $\operatorname{\mathcal{C}}$ also admits $K$-indexed limits and $U$ preserves $K$-indexed limits.
If $U$ is a left fibration and the $\infty $-category $\operatorname{\mathcal{D}}$ admits $K$-indexed colimits, then $\operatorname{\mathcal{C}}$ also admits $K$-indexed colimits and $U$ preserves $K$-indexed colimits.