Kerodon

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

Example 4.6.1.6. Let $\operatorname{\mathcal{C}}$ be a simplicial set containing vertices $X$ and $Y$. Let $K$ be a simplicial set, and let $\underline{X}, \underline{Y}: K \rightarrow \operatorname{\mathcal{C}}$ be the constant maps taking the values $X$ and $Y$, respectively. Then there is a canonical isomorphism of simplicial sets

\[ \operatorname{Hom}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) }( \underline{X}, \underline{Y} ) \simeq \operatorname{Fun}(K, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ). \]