Let $X$ be a compact Riemann surface. I would like to find a somehow complete reference for the proof of the so called non-Abelian Hodge correspondence relating Dolbeaut, Betti and Higgs bundle moduli spaces.

I've tried to read the original articles by Hitchin (1987) or Simpson (1990) but it seems to me that I've not found somehow a complete reference showing a precise proof of all the statements involved (for the case of a curve).

Does anyone know a book or notes related to this?