Definition 4.3.2.1 (Joins of Categories). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories. We define a category $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ as follows:
The set of objects $\operatorname{Ob}( \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}})$ is the disjoint union of $\operatorname{Ob}(\operatorname{\mathcal{C}})$ with $\operatorname{Ob}(\operatorname{\mathcal{D}})$.
Given a pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}})$, we have
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}}(X,Y) = \begin{cases} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) & \textnormal{if } X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}}) \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}(X,Y) & \textnormal{if } X,Y \in \operatorname{Ob}(\operatorname{\mathcal{D}}) \\ \ast & \textnormal{if } X \in \operatorname{Ob}(\operatorname{\mathcal{C}}), Y \in \operatorname{Ob}(\operatorname{\mathcal{D}}) \\ \emptyset & \textnormal{if } X \in \operatorname{Ob}(\operatorname{\mathcal{D}}), Y \in \operatorname{Ob}(\operatorname{\mathcal{C}}). \end{cases} \]Let $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ be morphisms in $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$. If $X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, then $g \circ f \in \operatorname{Hom}_{\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}}(X,Z)$ is given by the composition of morphisms in $\operatorname{\mathcal{C}}$. If $X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{D}})$, then $g \circ f$ is given by composition of morphisms in $\operatorname{\mathcal{D}}$. Otherwise, we let $g \circ f$ denote the unique morphism from $X$ to $Z$ (note that in this case, we necessarily have $X \in \operatorname{Ob}(\operatorname{\mathcal{C}})$ and $Z \in \operatorname{Ob}(\operatorname{\mathcal{D}})$).
We will refer to $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ as the join of $\operatorname{\mathcal{C}}$ with $\operatorname{\mathcal{D}}$.