Kerodon

Remark 5.6.2.15. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a diagram, and let $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ be the projection map. Let $\mathscr {F}^{\simeq }: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ denote the composition of $\mathscr {F}$ with the functor $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}^{\simeq }$ of Remark 5.5.4.9. It follows from Remark 5.6.2.14 that we can identify $\int _{\operatorname{\mathcal{C}}} \mathscr {F}^{\simeq }$ with the simplicial subset of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ given by those simplices $\Delta ^ n \rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ which carry each edge of $\Delta ^ n$ to a $U$-cocartesian morphism of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$. In other words, the projection map $U^{\simeq }: \int _{\operatorname{\mathcal{C}}} \mathscr {F}^{\simeq } \rightarrow \operatorname{\mathcal{C}}$ is the underlying left fibration of the cocartesian fibration $U$ (see Corollary 5.1.4.16).