Remark 5.2.3.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets, and let $K$ be a simplicial set. By virtue of Remark 4.3.3.21, morphisms from $K$ to the relative join $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ are given by maps $\pi : K \rightarrow \Delta ^1$ together with commutative diagrams
\[ \xymatrix@R =50pt@C=50pt{ \{ 0\} \times _{\Delta ^1} K \ar [d] \ar [r] & K \ar [d] & \{ 1\} \times _{\Delta ^1} K \ar [d] \ar [l] \\ \operatorname{\mathcal{C}}\ar [r]^-{F} & \operatorname{\mathcal{E}}& \operatorname{\mathcal{D}}. \ar [l]_-{G} } \]