Kerodon

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

Proof. We will show that $(1) \Rightarrow (2)$; the reverse implication follows from Corollary 5.6.1.16. Note that the map $\operatorname{N}_{\bullet }(U): \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a cocartesian fibration of simplicial sets (Example 5.1.4.2) and an inner covering map (Proposition 4.1.5.10). By virtue of Proposition 5.6.5.18, there exists a morphism of simplicial sets $\mathscr {F}': \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Pith}( \mathbf{Cat}) )$ and an isomorphism of simplicial sets $V: \int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F} \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}})$ which is compatible with $\operatorname{N}_{\bullet }(U)$. By virtue of Theorem 2.3.4.1 (and Corollary 2.3.4.5), we have $\mathscr {F}' = \operatorname{N}_{\bullet }^{\operatorname{D}}( \mathscr {F} )$ for a unique functor of $2$-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$. In this case, we can use Proposition 5.6.3.4 to identify $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F}'$ with the nerve of the ordinary category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$. Under this identification, $V$ corresponds to the nerve of an isomorphism $\int _{\operatorname{\mathcal{C}}} \mathscr {F}' \simeq \operatorname{\mathcal{E}}$ which is compatible with $U$. $\square$