Lemma 4.6.8.16. Let $K$ be a simplicial set and let $\operatorname{\mathcal{C}}$ be a simplicial category containing objects $X$ and $Y$. Then the natural map
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Functors $F: \operatorname{Path}[\Sigma (K)]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ with $F(x) = X$ and $F(y) = Y$} \} \ar [d] \\ \{ \textnormal{Morphisms $\Phi (K) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$} \} } \]
is a bijection.