# Kerodon

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Example 5.5.3.17. Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. For each object $C \in \operatorname{\mathcal{C}}$, the comparison map

$F: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \times _{ \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }})} \operatorname{N}_{\bullet }^{+}(\operatorname{Set_{\Delta }}) \rightarrow \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$

of Construction 5.5.3.16 induces an isomorphism of fibers

$F_{C}: \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }})} \operatorname{N}_{\bullet }^{+}(\operatorname{Set_{\Delta }}) \rightarrow \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \simeq \mathscr {F}(C)$

(see Example 5.5.3.7).