Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 5.5.3.5. Let $X$ be a Kan complex, which we regard as an object of the $\infty $-category $\operatorname{\mathcal{S}}$. Then Theorem 4.6.7.5 supplies a homotopy equivalence

\[ \theta _{X}: X = \operatorname{Hom}_{\operatorname{Kan}}(\Delta ^0, X)_{\bullet } \rightarrow \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{\mathcal{S}}}( \Delta ^0, X ) = \{ X\} \times _{\operatorname{\mathcal{S}}} \operatorname{\mathcal{S}}_{\ast }. \]

Beware that $\theta _{X}$ is generally not an isomorphism of simplicial sets.