# Kerodon

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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 $\operatorname{\mathcal{C}}= \operatorname{\mathcal{E}}[B]$. $\square$