Remark 4.3.4.4 (Symmetry). Let $X$ and $Y$ be topological spaces. Then there is a canonical homeomorphism $X \star Y \simeq Y \star X$, which is induced by the homeomorphism

\[ X \times [0,1] \times Y \rightarrow Y \times [0,1] \times X \quad \quad (x,t,y) \mapsto (y, 1-t, x). \]