Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 5.6.6.13 (Fibers of $V$). Let $\operatorname{\mathcal{C}}$ be a small $\infty $-category, which we regard as an object of the $\infty $-category $\operatorname{\mathcal{QC}}$. Then Construction 4.6.7.3 supplies a comparison map

\begin{eqnarray*} \operatorname{\mathcal{C}}& = & \operatorname{Hom}_{\operatorname{\mathbf{QCat}}}( \Delta ^0, \operatorname{\mathcal{C}}) \\ & \xrightarrow {\theta _{\operatorname{\mathcal{C}}}} & \operatorname{Hom}_{ \operatorname{ \pmb {\mathcal{QC}} }}^{\mathrm{L}}( \Delta ^0, \operatorname{\mathcal{C}}) \\ & = & \{ \operatorname{\mathcal{C}}\} \times _{ \operatorname{ \pmb {\mathcal{QC}} }} \operatorname{ \pmb {\mathcal{QC}} }_{\operatorname{Obj}} \\ & = & \{ \operatorname{\mathcal{C}}\} \times _{ \operatorname{\mathcal{QC}}} \operatorname{\mathcal{QC}}_{\operatorname{Obj}}, \end{eqnarray*}

which is an equivalence of $\infty $-categories (Theorem 4.6.7.9). Beware that $\theta _{\operatorname{\mathcal{C}}}$ is generally not an isomorphism of simplicial sets (though it is bijective on $n$-simplices for $n \leq 1$; see Example 5.6.6.12).