Let $m$ and $n$ be two non-zero natural integers that are coprime. Show that the map
$$\pi : \left\lvert\, \begin{gathered} \mathcal{D}_n \times \mathcal{D}_m \rightarrow \mathcal{D}_{mn} \\ \left( d_1, d_2 \right) \mapsto d_1 d_2 \end{gathered} \right.$$
is well-defined and realizes a bijection between $\mathcal{D}_n \times \mathcal{D}_m$ and $\mathcal{D}_{mn}$.