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Proposition 5.5.4.2. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a morphism of simplicial sets. Then the projection map $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration of simplicial sets.

Proof. By construction, the morphism $U$ fits into a pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} \ar [r] \ar [d]^-{U} & \operatorname{\mathcal{QC}}^{\operatorname{lax}}_{\ast } \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{\mathscr {F}} & \operatorname{\mathcal{QC}}, } \]

where

\[ \operatorname{\mathcal{QC}}^{\operatorname{lax}}_{\ast } = \operatorname{\mathcal{QC}}\times _{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }})} \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }})_{ \Delta ^{0} / } \]

is the $\infty $-category introduced in Construction 5.4.6.10. It will therefore suffice to show that the projection map $\operatorname{\mathcal{QC}}^{\operatorname{lax}}_{\ast } \rightarrow \operatorname{\mathcal{QC}}_{\ast }$ is a cocartesian fibration of simplicial sets, which follows from Proposition 5.4.6.11. $\square$