# Kerodon

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Remark 4.6.5.3. Let $\operatorname{\mathcal{C}}$ be a simplicial set containing vertices $X$ and $Y$, which we also regard as vertices of the opposite simplicial set $\operatorname{\mathcal{C}}^{\operatorname{op}}$. Then we have canonical isomorphisms of simplicial sets

$\operatorname{Hom}_{\operatorname{\mathcal{C}}^{\operatorname{op}}}^{\mathrm{L}}(X,Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(Y,X)^{\operatorname{op}} \quad \quad \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\operatorname{op}}}^{\mathrm{R}}(X,Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(Y,X)^{\operatorname{op}}.$