Corollary 8.4.5.10 (05JD)

Corollary 8.4.5.10. Let $\mathbb {K}$ be a collection of simplicial sets and let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$. Let $U: \operatorname{\mathcal{E}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a left fibration of $\infty $-categories. The following conditions are equivalent:

$(1)$: The covariant transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\widehat{\operatorname{\mathcal{C}}}}: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{S}}$ preserves $K$-indexed colimits, for each $K \in \mathbb {K}$.
$(2)$: The projection map $\operatorname{\mathcal{E}}\times _{ \widehat{\operatorname{\mathcal{C}}} } \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is left cofinal.

Proof. It follows from Proposition 8.4.5.3 that the functor $h$ is fully faithful. We can therefore replace $\operatorname{\mathcal{C}}$ by its essential image and thereby reduce to the case where $\operatorname{\mathcal{C}}$ is a replete full subcategory of $\widehat{\operatorname{\mathcal{C}}}$ (and $h$ is the inclusion functor). In this case, condition $(1)$ is equivalent to the requirement that the functor $\operatorname{Tr}_{\operatorname{\mathcal{E}}/ \widehat{\operatorname{\mathcal{C}}} }$ is left Kan extended from $\operatorname{\mathcal{C}}$ (Lemma 8.4.5.9). The equivalence $(1) \Leftrightarrow (2)$ is now a special case of Corollary 7.4.3.15. $\square$