Kerodon

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Remark 5.6.2.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. A commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r]^-{ \widetilde{\mathscr {F}} } & \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [d]^{V} \\ \operatorname{\mathcal{C}}\ar [r]^-{ \mathscr {F} } & \operatorname{\mathcal{QC}}} \]

witnesses $\mathscr {F}$ as a covariant transport representation of $U$ if and only if it satisfies the following pair of conditions:

$(a)$

For every vertex $C \in \operatorname{\mathcal{C}}$, the map of fibers

\[ \widetilde{\mathscr {F}}_{C}: \operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \{ C\} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F} \]

is an equivalence of $\infty $-categories.

$(b)$

The morphism $\widetilde{\mathscr {F}}$ carries $U$-cocartesian edges of $\operatorname{\mathcal{E}}$ to $V$-cocartesian edges of $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$.

See Proposition 5.1.6.14. Moreover, we can replace $(b)$ by the following a priori weaker condition (see Remark 5.1.5.8):

$(b')$

For every vertex $X \in \operatorname{\mathcal{E}}$ and every edge $\overline{e}: U(X) \rightarrow \overline{Y}$ in $\operatorname{\mathcal{C}}$, there exists a $U$-cocartesian edge $e: X \rightarrow Y$ of $\operatorname{\mathcal{E}}$ for which $U( e) = \overline{e}$ and and $\widetilde{\mathscr {F}}(e)$ is a $V$-cocartesian edge of $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$.