Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 7.6.2.18. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an essentially small left fibration of $\infty $-categories. Then a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ 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 $U$.

Proof. Combine Proposition 7.6.2.17 with the equivalence $\operatorname{\mathcal{S}}_{\ast } \hookrightarrow \{ \Delta ^0 \} \operatorname{\vec{\times }}_{\operatorname{\mathcal{S}}} \operatorname{\mathcal{S}}$ of Theorem 4.6.4.17. $\square$