The covering space of a region contained in complex plane delete two points.

Question

We all know that C \ {0,1} can be given the Poincare hyperbolic metric, so that a region W in it is an embedded manifold of negative constant curvature. Hence the covering space of W is a hyperbolic space i.e., the Poincare disc. But I want to show that the covering map is actually locally biholomorphic in order to prove the Picard's theorem. Any help is appreciable, thanks a lot.