Every proper simply connected plane domain is uniquely normalized to the unit disc by fixing one point and a positive derivative.Suggest a slogan
Let \(\Omega\subsetneq\CC\) be a nonempty simply connected open set. Then for every \(z_0\in\Omega\) there exists a unique biholomorphism \(F\colon\Omega\to\DD\) such that \(F(z_0)=0\) and \(F'(z_0)>0\). In particular, any two nonempty simply connected open sets \(\Omega_1,\Omega_2\subsetneq\CC\) are biholomorphic.
This is [@SS03].