Corollary 7.4.5.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration between small simplicial sets and let $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a covariant transport representation for $U$. Then the diagram $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ admits a colimit in the $\infty $-category $\operatorname{\mathcal{S}}$. Moreover, a Kan complex $X$ is a colimit of $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ if and only if there exists a weak homotopy equivalence $\operatorname{\mathcal{E}}\rightarrow X$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Since $U$ is a left fibration, every edge of $\operatorname{\mathcal{E}}$ is $U$-cocartesian (Example 5.1.1.3). Let $W$ be the collection of all $U$-cocartesian edges of $\operatorname{\mathcal{E}}$. By virtue of Corollary 7.4.3.11, an $\infty $-category $X$ is a colimit of $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ in the $\infty $-category $\operatorname{\mathcal{QC}}$ if and only if there exists a functor $f: \operatorname{\mathcal{E}}\rightarrow X$ which exhibits $X$ as a localization of $\operatorname{\mathcal{E}}$ with respect to $W$. By virtue of Proposition 6.3.1.20, this is equivalent to the requirement that $X$ is a Kan complex and that $f$ is a weak homotopy equivalence. In this case, $X$ is also a colimit of the diagram $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ in the full subcategory $\operatorname{\mathcal{S}}\subseteq \operatorname{\mathcal{QC}}$ (Proposition 7.4.5.1). $\square$