Kerodon

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

Proposition 5.4.3.8. The coslice comparison functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } ) \rightarrow \operatorname{\mathcal{S}}_{\ast }$ is an equivalence of $\infty $-categories.

Proof. Note that, for every pair of pointed Kan complexes $(X,x)$ and $(Y,y)$, the evaluation map $\operatorname{Fun}(X,Y) \rightarrow \operatorname{Fun}( \{ x\} , Y)$ is a Kan fibration (Corollary 3.1.3.3). Proposition 5.4.3.8 is therefore a special case of Theorem 5.4.2.21. $\square$