Kerodon

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

Corollary 5.3.4.19. Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories indexed by $\operatorname{\mathcal{C}}$, and let $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be the cocartesian fibration of Proposition 5.3.3.15. Then there exists a levelwise categorical equivalence from $\mathscr {F}$ to the strict transport representation $\operatorname{sTr}_{ \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}}$.

Proof. Combine Proposition 5.3.4.17 with Remark 5.3.4.10 (for a more precise statement, see Construction 7.5.3.3). $\square$