Remark 4.3.2.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}'$ be functors. Then $F$ and $G$ induce a functor
\[ (F \star G): \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}' \star \operatorname{\mathcal{D}}', \]
which is uniquely determined by the requirement that it coincides with $F$ on the full subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ and with $G$ on the full subcategory $\operatorname{\mathcal{D}}\subseteq \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$. We can therefore regard the join construction as a functor
\[ \star : \operatorname{Cat}\times \operatorname{Cat}\rightarrow \operatorname{Cat}\quad \quad (\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \mapsto \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}, \]
where $\operatorname{Cat}$ denotes the category of (small) categories.