Kerodon

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

Remark 9.5.0.4. Let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$, which we identify with vertices of the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$. Using Example 9.5.0.3, we can identify $n$-simplices $\sigma $ of the simplicial set $\operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}( X, Y)$ with commutative diagrams . Allowing $[n] \in \operatorname{{\bf \Delta }}$ to vary, we obtain canonical isomorphisms of simplicial sets

\[ \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Y) \simeq \operatorname{N}_{\bullet }( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ) \quad \quad \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}^{\mathrm{R}}( X, Y) \simeq \operatorname{N}_{\bullet }( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) )^{\operatorname{op}}. \]