Example 4.6.1.6. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, so that the join $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ is also an $\infty $-category (Corollary 4.3.3.25). Then the morphism spaces in $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ are described by the formula
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}}(X,Y) \simeq \begin{cases} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) & \textnormal{if } X,Y \in \operatorname{\mathcal{C}}\\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}(X,Y) & \textnormal{if } X,Y \in \operatorname{\mathcal{D}}\\ \Delta ^{0} & \textnormal{if } X \in \operatorname{\mathcal{C}}, Y \in \operatorname{\mathcal{D}}\\ \emptyset & \textnormal{if } X \in \operatorname{\mathcal{D}}, Y \in \operatorname{\mathcal{C}}. \end{cases} \]