Proposition 7.4.3.6. Let $\kappa $ be an uncountable cardinal and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be the covariant transport representation of a left fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, where $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small. Then a Kan complex $X$ is a colimit of $\mathscr {F}$ (in the $\infty $-category $\operatorname{\mathcal{S}}^{< \kappa }$) if and only if there exists a weak homotopy equivalence $\operatorname{\mathcal{E}}\rightarrow X$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Using Proposition 5.6.7.2, we can reduce to the situation where $\operatorname{\mathcal{C}}$ is an $\infty $-category. In this case, Corollary 7.4.2.15 guarantees that $\mathscr {F}$ is a left Kan extension of the constant functor $\underline{ \Delta ^{0} }_{\operatorname{\mathcal{E}}}$ along $U$. Applying Corollary 7.3.8.20, we see that an object $X \in \operatorname{\mathcal{S}}^{< \kappa }$ is a colimit of $\mathscr {F}$ if and only if it is a colimit of the constant diagram $\underline{ \Delta ^{0} }_{\operatorname{\mathcal{E}}}$. By virtue of Variant 7.1.2.11, this is equivalent to the existence of a weak homotopy equivalence $\operatorname{\mathcal{E}}\rightarrow X$. $\square$