automorphisms of unit disk

All automorphisms of the complex unit disk to itself,can be written in the form $f_a(z)=e^{i\theta }\frac{z-a}{1-\overline{a}z}$ where $a\in \mathrm{\Delta }$ and $\theta \in S^1$.

This map sends $a$ to $0$, $1/\overline{a}$ to $\mathrm{\infty }$ and the unit circle to the unit circle.