Corollary 7.1.7.5. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories and let $K$ be a weakly contractible simplicial set. Then:
If $U$ is a left fibration and the $\infty $-category $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits, then the $\infty $-category $\operatorname{\mathcal{E}}$ also admits $K$-indexed colimits and the functor $U$ preserves $K$-indexed colimits.
If $U$ is a right fibration and the $\infty $-category $\operatorname{\mathcal{C}}$ admits $K$-indexed limits, then the $\infty $-category $\operatorname{\mathcal{E}}$ also admits $K$-indexed limits and the functor $U$ preserves $K$-indexed limits.