Kerodon

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

Corollary 5.5.4.22. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a functor of $\infty $-categories and set $\operatorname{\mathcal{E}}= \int _{\operatorname{\mathcal{C}}} \mathscr {F}$. Then there is a canonical isomorphism $\mathrm{h} \mathit{\mathscr {F}} \xrightarrow {\sim } \operatorname{hTr}_{\operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}}$ in the functor category $\operatorname{Fun}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \mathrm{h} \mathit{\operatorname{\mathcal{QC}}})$, which carries each vertex $C \in \operatorname{\mathcal{C}}$ 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 5.5.4.18.