 Solution

A conformal map preserves the magnitude and orientation of angles between intersecting smooth curves locally. For a complex holomorphic function, the condition at $z_0$ is $f^{\prime }(z_0)\ne 0$: the first-order change $\delta w=f^{\prime }(z_0)\delta z$ rotates and uniformly scales every tangent vector by the same complex factor. This local statement does not require a globally one-to-one map.