Kerodon

Corollary 7.4.2.15. Let $\kappa $ be an uncountable cardinal and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of $\infty $-categories which is essentially $\kappa $-small. Then a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ is a covariant transport representation for $U$ (in the sense of Definition 5.6.5.1) if and only if it is a left Kan extension of the constant functor $\underline{ \Delta ^0 }_{\operatorname{\mathcal{E}}}$ along $U$.