Kerodon

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

Corollary 4.6.7.19. Let $A$ and $B$ be simplicial sets, and let $\operatorname{\mathcal{E}}[B]$ be the simplicial category of Notation 4.6.7.1. Then we have a canonical bijection

\[ \{ \textnormal{Morphisms $A \rightarrow \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}}[B])}^{\mathrm{L}}(x,y)$} \} \simeq \{ \textnormal{Morphisms $\Phi (A) \rightarrow B$} \} . \]

Proof. Apply Corollary 4.6.7.17 in the special case $\operatorname{\mathcal{C}}= \operatorname{\mathcal{E}}[B]$. $\square$