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Proposition 5.6.2.19. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $\mathscr {F}, \mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be diagrams, and let $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ and $U': \int _{\operatorname{\mathcal{C}}} \mathscr {F}' \rightarrow \operatorname{\mathcal{C}}$ be the projection maps. If $\mathscr {F}$ and $\mathscr {F}'$ are isomorphic as objects of the diagram $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})$, then $U$ and $U'$ are equivalent as cocartesian fibrations over $\operatorname{\mathcal{C}}$ (in the sense of Definition 5.1.6.1).

Proof. Apply Proposition 5.1.6.10 to the cocartesian fibration $\operatorname{\mathcal{QC}}_{\operatorname{Obj}} \rightarrow \operatorname{\mathcal{QC}}$ of Proposition 5.5.6.11. $\square$