Complex diffeomorphism and a domain being Polygon

Question

I've been studying the lecture notes of 246A taught by Prof. Terence Tao.

• https://terrytao.wordpress.com/2016/10/18/246a-notes-5-conformal-mapping/#more-9481

But I am stuck at understanding this sentence,

in particular, such diffeomorphisms exist if (the domain of such diffeomorphism)U is a polygon

** Line 3 of Section 4. Schwarz-Christoffel mappings

Could anyone tell me why so?

Answer 1

The polygon $U$ is simply connected and omits at least one point, so there is a bijective Riemann map from the disk to $U$.