Kerodon

Remark 5.6.2.14 (Cocartesian Morphisms of the $\infty $-Category of Elements). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a morphism of simplicial sets, so that the projection map $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration of simplicial sets (Proposition 5.6.2.2). Then an edge $(f,u): (C,X) \rightarrow (D,Y)$ of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is $U$-cocartesian if and only if $u: \mathscr {F}(f)(X) \rightarrow Y$ is an isomorphism in the $\infty $-category $\mathscr {F}(D)$ (see Example 5.5.6.12).