Existence of conformal equivalence between doubly connected domain and annulus

The following math overflow post

https://mathoverflow.net/questions/261535/mapping-the-doubly-connected-domain-to-an-annulus

provides a sketch proof of the fact that any (non-degenerate) doubly connected domain is conformally equivalent to an annulus (with outer radius 1).

To construct the holomorphic bijection the proof apparently uses (in the last lines) the logarithm as a multivalued function ( no holomorphic branch can be defined on the doubly connected region).

I can convince myself that the candidate conformal equivalence is indeed a function but find it difficult to verify holomorphy ( as one is composing with a set valued function). Any ideas on how to proceed? Many thanks!