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Remark 5.7.2.17. Let $U: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ and $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }})$ be morphisms of simplicial sets, and let $\mathscr {F}'$ denote the composition $(\mathscr {F}' \circ U): \operatorname{\mathcal{C}}' \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }})$. Then the simplicial set $\int _{\operatorname{\mathcal{C}}'} \mathscr {F}'$ can be identified with the fiber product $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F}$.