Hyperbolic structure on surface gives a complex structure

Question

My question is from A primer on mapping class group, p.295:

I can see $X=\Delta/\Gamma$ has an induced hyperbolic structure, but why conversely any such hyperbolic structure gives a complex structure on $X$? Also for the bijection, how do we know these induced maps are inverse to each other?

Answer 1

The key fact which answers both of your questions is that the group of orientation preserving hyperbolic isometries of $\Delta$ is identical to the group of biholomorphic automorphisms of $\Delta$.

So, just as any Riemann surface structure on $S_g$ is the quotient of $\Delta$ by a group of bilomorphic automorphisms and hence has an induced hyperbolic structure, similarly any hyperbolic structure on $S_g$ is the quotient of $\Delta$ by a group of orientation preserving hyperbolic isometries and hence has an induced Riemann surface structure.