Let $$V := \left \{ z: \Im(z) > 0 \right \} $$ be the upper half-plane. For $a \in V$ we define $$h_{a}(z) = \frac{z-a}{z-\bar{a}} $$ for $ z \neq \bar{a}$.
So that $h_{a}$ is a conformal map $V \rightarrow D(0,1)$. Let now $f: D(0,1) \rightarrow V$ be a conformal function such that $f(0)=a$. Show that there is a complex number $c$ with $|c|=1$ such that $$f(z) = \frac{ac-\bar{a}z}{c-z} $$ for all $z \in D(0,1)$.
It says also we should consider the function $ f \circ h_{a}^{-1} $ and find a general expression for conformal maps $ V \rightarrow V$.