Remark 4.6.8.2. The simplicial category $\operatorname{\mathcal{E}}[K]$ is characterized by the following universal property: if $\operatorname{\mathcal{C}}$ is any simplicial category containing a pair of objects $X$ and $Y$, then the natural map
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Simplicial functors $F: \operatorname{\mathcal{E}}[K] \rightarrow \operatorname{\mathcal{C}}$ with $F(x) = X$ and $F(y) = Y$} \} \ar [d] \\ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(K, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } ) } \]
is a bijection (see Proposition 2.4.5.9).