Proposition 7.4.3.1. Let $\operatorname{\mathcal{C}}$ be a small simplicial set and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be the covariant transport representation of a left fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ (Definition 5.6.5.1). Then an object $X \in \operatorname{\mathcal{S}}$ is a colimit of $\mathscr {F}$ if and only if there exists a weak homotopy equivalence $\operatorname{\mathcal{E}}\rightarrow X$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$