# Question about conformally equivalent domains.

This is the well known definition of conformal equivallence between domains in the complex plane:

Let $U$ and $U^{\prime}$ be two domains in the complex plane. We say $U$ and $U^{\prime}$ are conformally equivalent if there is a map $f$ between them such that $f$ is holomorphic and inyective (a conformal map).

My question is: having one such map between these domains, say $f: U \rightarrow U^{\prime},$ implies the existence of a conformal map in the other direction: $g: U^{\prime} \rightarrow U?$ And if so, what theorem guarantees this?