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Example 5.5.4.18 (Fibers of the $\infty $-Category of Elements). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{ \pmb {\mathcal{QC}} }$ be a morphism of simplicial sets. For each vertex $C \in \operatorname{\mathcal{C}}$, Remark 5.5.4.17 and Example 5.5.4.16 supply a canonical isomorphism

\[ \{ C\} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F} \simeq \operatorname{Hom}^{\mathrm{L}}_{\operatorname{ \pmb {\mathcal{QC}} }}( \Delta ^{0}, \mathscr {F}(C) ). \]

In particular, Construction 4.6.6.3 supplies a comparison functor $\theta _{C}: \mathscr {F}(C) \rightarrow \{ C \} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ which is an equivalence of $\infty $-categories (Theorem 4.6.6.9), but generally not an isomorphism of simplicial sets.