Remark 4.3.2.6. Let $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ be categories. Then there is a canonical isomorphism of iterated joins
\[ \alpha : \operatorname{\mathcal{C}}\star (\operatorname{\mathcal{D}}\star \operatorname{\mathcal{E}}) \simeq (\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}) \star \operatorname{\mathcal{E}}, \]
characterized by the requirement that it restricts to the identity on $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ (which we can regard as full subcategories of both $\operatorname{\mathcal{C}}\star (\operatorname{\mathcal{D}}\star \operatorname{\mathcal{E}})$ and $(\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}) \star \operatorname{\mathcal{E}}$, by means of Remark 4.3.2.2).