About the uniqueness part of the Riemann Mapping Theorem

Question

I was just reading some preliminary fact for the Riemann Mapping Theorem and there is a Lemma of Poincare given which i think is supposed to set up the uniqueness part. It is stated as follow.

Let Ω ⊂ C be an open set, and let α ∈ Ω. Assume that φ : Ω → D and ψ : Ω→D are biholomorphic mappings onto D such that φ(α) = ψ(α) = 0 and that both φ'(α) and ψ'(α) are positive real numbers. Then φ = ψ.

There is no proof given so i tried myself but i haven't had useful ideas. I only know figured that you have to use Schwarz's Lemma somehow but i couldn't introduce a fitting Map for this matter. could you help me out

Answer 1

A hint:

Consider the map $$f:=\phi\circ\psi^{-1}:\quad D\to D\ ,$$ and look at $f(0)$, $f'(0)$.