Exercise states:
Let $D_1, D_2$ be two simply-connected open sets, both not equal to $\mathbb{C}$. Prove that for every $z_0 \in D_1, w_0 \in D_2$ there exists a single biholomorphic function $f : D_1 \to D_2$ such that $$f(z_0)=w_0,\ \ \ \ \ \ \arg(f'(z_0))=\frac{\pi}{2}$$
I managed to prove existence: by the Riemann mapping theorem, there exists a mapping $g_1: D_1 \to \Delta$, $\Delta$ is the open unit circle, such that $$g_1(z_0)=0,\ \ \ \ \ \ \arg(g_1'(z_0))=\frac{\pi}{2}$$ and also there exists a function $g_2 : D_2 \to \Delta$, that satisfies $$g_2(w_0)=0,\ \ \ \ \ \ \arg(g_2'(w_0))=0$$
If we define $f := g_2^{-1} \circ g_1$, then $f(z_0)=w_0$ and
$\arg(f'(z_0))=\frac{\pi}{2}$. But I didn't find a way to prove uniqueness. What's the way to prove it?