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Example (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 Combining Example with Remark, 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 and Example, it corresponds to the identity morphism $\operatorname{id}: \mathscr {F}(C) \rightarrow \mathscr {F}(C)$).