Proposition 5.6.2.22 (02S8)

Proposition 5.6.2.22. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a functor of $\infty $-categories, let $\operatorname{\mathcal{E}}= \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ denote the $\infty $-category of elements of $\mathscr {F}$, and let

\[ \operatorname{hTr}_{ \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{QC}}} \]

denote the enriched homotopy transport representation associated to the cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ (see Construction 5.2.8.9). Then there is a canonical isomorphism of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functors $\theta : \mathrm{h} \mathit{\mathscr {F}} \rightarrow \operatorname{hTr}_{ \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}}$, which carries each object $C \in \operatorname{\mathcal{C}}$ to the comparison map

\[ \theta _{C}: \mathscr {F}(C) \rightarrow \operatorname{hTr}_{\operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}}(C) = \{ C\} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F} \]

of Example 5.6.2.19.

Proof. By virtue of Remarks 5.2.8.10 and 5.6.2.18, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \operatorname{\mathcal{QC}}$ and that $\mathscr {F}$ is the identity functor. In this case, the desired result is a restatement of Proposition 5.5.6.14. $\square$