How to prove that the Douady-Hubbard conformal map from the exterior of Mandelbrot Set to exterior of unit disc is actually holomorphic? I was reading the Orsay Notes on Exploring the Mandelbrot Set. (https://pi.math.cornell.edu/~hubbard/OrsayEnglish.pdf)
On Page 64, it is proven that the Mandelbrot Set is connected.
I understood the rest of the proof but I can't understand the part where they prove that $\Phi:\mathbb{C}\backslash M\mapsto \mathbb{C}\backslash \mathbb{D}$, given by $\Phi(c)=\phi_c(c)$ is holomorphic. It is just stated that $(z,c)\to \phi_c(z)$ is holomorphic in two variables ($z\in\mathbb{C}:G_c(z)>G_c(0))$. I don't understand how is this proved and how does this even imply the holmorphicity of $\Phi(c)=\phi_c(c)$.
(The notations are as used in the book).

To prove holomorphicity in two variables, you just need to prove continuity and then holomorphicity with respect to each variable separately. Just go through the construction of $\phi$ and check it (it is constructed as a locally uniform limit of holomorphic functions).
For the second question how this implies holomorphicity of $\Phi$, this is just the fact that $\Phi$ is the composition of the two holomorphic maps $c \mapsto (c,c)$ and $(z,c) \mapsto \phi_c(z)$.
