Lemma 8.6.5.15. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories, let $\kappa $ be an uncountable cardinal such that each fiber of $U$ is locally $\kappa $-small. Let $\widetilde{e}$ be an edge of the simplicial set $\operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$ corresponding to a pair $(e, \mathscr {F} )$, where $e: C \rightarrow D$ is an edge of $\operatorname{\mathcal{C}}$ and $\mathscr {F}: \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ is a functor. The following conditions are equivalent:
- $(1)$
The edge $\widetilde{e}$ is $\pi ^{\operatorname{corep}}$-cocartesian (where $\pi ^{\operatorname{corep}}: \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa }) \rightarrow \operatorname{\mathcal{C}}$ denotes the projection map).
- $(2)$
There exists an object $X \in \operatorname{\mathcal{E}}_{C}$, a vertex $\eta \in \mathscr {F}(X)$ which exhibits $\mathscr {F}|_{ \operatorname{\mathcal{E}}_{C} }$ as corepresented by the object $X$, and a $U$-cocartesian morphism $\overline{e}: X \rightarrow Y$ such that $U( \overline{e} ) = e$ and the vertex $\mathscr {F}( \overline{e} )( \eta ) \in \mathscr {F}(Y)$ exhibits $\mathscr {F}|_{ \operatorname{\mathcal{E}}_{D} }$ as corepresented by the object $Y$.
- $(3)$
For every object $X \in \operatorname{\mathcal{E}}_{C}$, every vertex $\eta \in \mathscr {F}(X)$ which exhibits $\mathscr {F}|_{\operatorname{\mathcal{E}}_{C}}$ as corepresented by the object $X$, and every $U$-cocartesian morphism $\overline{e}: X \rightarrow Y$ satisfying $U( \overline{e} ) = e$, the vertex $\mathscr {F}( \overline{e} )( \eta ) \in \mathscr {F}(Y)$ exhibits $\mathscr {F}|_{ \operatorname{\mathcal{E}}_{D} }$ as corepresented by the object $Y$.