Remark 11.5.0.67. We will see later that if $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration of $\infty $-categories, then the homotopy transport representation $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ can be refined to a functor of $\infty $-categories $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$, which we will refer to as the covariant transport representation of $U$ (see Definition 5.6.5.1). Moreover, the refined analogues of Questions 11.5.0.59 and Question 5.2.0.7 both have positive answers:
A cocartesian fibration of $\infty $-categories $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ can be recovered (up to equivalence) from the transport representation $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$.
Every functor of $\infty $-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ can be obtained (up to isomorphism) as the transport representation of a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. Moreover, there is an explicit realization of $\operatorname{\mathcal{E}}$ as the $\infty $-category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ (see Definition 5.6.2.1).
From this perspective, the negative answers to Questions 11.5.0.59 and 5.2.0.7 are due to the fact that a functor of ordinary categories $\mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ cannot generally be lifted to a functor of $\infty $-categories $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$, and that such a lifting need not be unique when it exists.