For any $z_0$ in the unit disk $\mathbb{D}$, the Möbius map $\varphi_{z_0}: z \mapsto \frac{z_0 - z}{1 - z\bar{z_0}}$ is a conformal map...

Question

...from $\mathbb{D}$ to $\mathbb{D}$.

In the lecture our instructor shows several properties of the map $\varphi_{z_0}$, such as it maps $\mathbb{D}$ to $\mathbb{D}$, it is onto and injective, and $\varphi'_{z_0}(z) \neq 0$ for $z \in \mathbb{D}$, and that concludes the proof. What about the requirement that the differential $d\varphi_{z_0}(z)$ be angle-preserving?