Kerodon

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

Remark 4.6.7.10. Let $\operatorname{\mathcal{C}}$ be a simplicial category containing a pair of objects $X$ and $Y$, and suppose that the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is an $\infty $-category. Applying Theorem 4.6.7.9 to the opposite simplicial category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ (and using Remark 4.6.5.3), we obtain an equivalence of $\infty $-categories

\[ \theta ': \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }^{\operatorname{op}} \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}^{\mathrm{R}}( X, Y), \]

which can be described explicitly using a variant of Construction 4.6.7.3.