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Example 5.3.4.15 (Comparison of Fibers). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets and let $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ be the morphism of Construction 5.3.4.11. Combining Example 5.3.4.13 with Remark 5.3.4.14, we see that for every object $C \in \operatorname{\mathcal{C}}$, the induced map of fibers

$\{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \rightarrow \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$

is an isomorphism of simplicial sets (under the identifications provided by Remark 5.3.2.3 and Example 5.3.3.8, it corresponds to the identity morphism $\operatorname{id}: \mathscr {F}(C) \rightarrow \mathscr {F}(C)$).