Variant 7.4.2.14. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of $\infty $-categories and let $\kappa $ be an uncountable cardinal. Suppose we are given a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ and a natural transformation $\alpha : \underline{ \Delta ^{0} }_{\operatorname{\mathcal{E}}} \rightarrow \mathscr {F}|_{\operatorname{\mathcal{E}}}$. If $U$ is essentially $\kappa $-small, then the following conditions are equivalent:
- $(1)$
The natural transformation $\alpha $ exhibits $\mathscr {F}$ as a covariant transport representation for $U$.
- $(2)$
The natural transformation $\alpha $ exhibits $\mathscr {F}$ as a left Kan extension of the constant functor $\underline{ \Delta ^{0} }|_{\operatorname{\mathcal{E}}}$ along $U$.
Beware that, if $U$ is not assumed to be essentially $\kappa $-small, then it is possible for condition $(2)$ to be satisfied while condition $(1)$ is not (see Exercise 7.1.2.12).