My work:
let $M_1 = J \frac{z-m}{1-\bar m z}, M_2 = K \frac{z-n}{1-\bar n z} $ where $|J|,|K| = 1$ .$n,m \in {\mathscr d} $ then I simplified a bit and got $M_1 \circ M_2 = JK \frac {(z-n) - m(1-\bar n z)}{1 - \bar n z - \bar m k (z-n)} $. not sure how to continue from here