Corollary 7.4.1.21. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of $\infty $-categories, let $K$ be a simplicial set, and suppose that $\operatorname{\mathcal{C}}$ admits $K$-indexed limits. The following conditions are equivalent:
- $(1)$
The covariant transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ preserves $K$-indexed limits.
- $(2)$
The $\infty $-category $\operatorname{\mathcal{E}}$ admits $K$-indexed limits which are preserved by the functor $U$.