Kerodon

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

Proposition 2.4.6.8. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a locally Kan simplicial category. Then the construction above induces an isomorphism of categories $U: \mathrm{h} \mathit{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}) } \xrightarrow {\sim } \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.

Proof of Proposition 2.4.6.8. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a locally Kan simplicial category; we wish to show that the comparison map $U: \mathrm{h} \mathit{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) } \xrightarrow {\sim } \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is an isomorphism of categories. By construction, $U$ is bijective on objects. It will therefore suffice to show that for every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, the induced map

\[ U_{X,Y}: \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}}( X, Y) \rightarrow \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }(X, Y) \]

is a bijection. This is precisely the content of Example 2.4.6.10. $\square$