Remark 4.6.8.13. Let $K$ be a simplicial set, and let $\operatorname{\mathcal{D}}$ be another simplicial set containing vertices $X$ and $Y$. Unwinding the definitions, we have a canonical bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Morphisms $K \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}^{\mathrm{L}}(X,Y)$} \} \ar [d]^{\sim } \\ \{ \textnormal{Morphisms $F: \Sigma (K) \rightarrow \operatorname{\mathcal{D}}$ with $F(x) = X$ and $F(y) = Y$} \} . } \]