Lemma 9.4.2.2 (05LF)

Lemma 9.4.2.2. Let $\mathbb {K}$ be a collection of simplicial sets and let $\widehat{U}: \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration. Then $\widehat{U}$ is $\mathbb {K}$-cocomplete (in the sense of Variant 9.4.1.4) if and only if it satisfies the following conditions:

For each object $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\widehat{\operatorname{\mathcal{E}}}_ C = \{ C\} \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}$ is $\mathbb {K}$-cocomplete.
For each edge $f: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the covariant transport functor $f_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C'}$ preserves $K$-indexed colimits for each $K \in \mathbb {K}$.

Proof. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}$ is an $\infty $-category (or even that $\operatorname{\mathcal{C}}= \Delta ^1$). In this case, the desired result follows from the characterization of $\widehat{U}$-colimit diagrams supplied by Proposition 7.3.9.2. $\square$