Remark 5.2.3.14 (024F): Morphism Spaces in the Relative Join

Remark 5.2.3.14 (Morphism Spaces in the Relative Join). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets. If $X$ and $Y$ are vertices of the relative join $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$, then we have canonical isomorphisms of simplicial sets

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}}( X, Y) \simeq \begin{cases} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) & \textnormal{ if $X,Y \in \operatorname{\mathcal{C}}$} \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( X,Y) & \textnormal{ if $X,Y \in \operatorname{\mathcal{D}}$} \\ \operatorname{Hom}_{\operatorname{\mathcal{E}}}( F(X), G(Y) ) & \textnormal{ if $X \in \operatorname{\mathcal{C}}$, $Y \in \operatorname{\mathcal{D}}$} \\ \emptyset & \textnormal{ if $X \in \operatorname{\mathcal{D}}$, $Y \in \operatorname{\mathcal{C}}.$} \end{cases} \]

The pinched morphism spaces $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}}( X, Y)$ and $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}}( X, Y)$ admit similar descriptions.