Definition 5.6.2.1. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }})$ be a morphism of simplicial sets. We let $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ denote the fiber product $\operatorname{\mathcal{C}}\times _{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }})_{ \Delta ^{0} / }$, so that we have a pullback diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} \ar [r] \ar [d]^-{U} & \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }})_{\Delta ^{0}/} \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{\mathscr {F}} & \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }}). } \]
We will refer to $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ as the projection map.