# Kerodon

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

Example 5.6.5.7 (Strict Transport). Let $\operatorname{\mathcal{C}}$ be an ordinary category, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a cocartesian fibration of $\infty$-categories, and let $\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be the strict transport representation of $U$ (Construction 5.3.1.5). Then the functor

$\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}) = \operatorname{\mathcal{QC}}$

is a covariant transport representation for $U$ (in the sense of Definition 5.6.5.1). In other words, $U$ is equivalent to the cocartesian fibration $U': \int _{\operatorname{\mathcal{C}}} \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. To see this, we observe that both $U$ and $U'$ are equivalent to the cocartesian fibration $\operatorname{N}_{\bullet }^{ \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$: this follows from Theorem 5.3.5.6 and Proposition 5.6.4.8.