I'm trying to prove if $f:X\rightarrow Y$ is a local homeomorphism with $Y$ a Riemann surface then there is a unique complex structure on $X$ such that $f$ is holomorphic.
I've proved the existence part and for uniqueness I suppose that $\mathcal{A}=\{(U_x, \phi_x)\}$, $\mathcal{C}=\{(W_a,\chi_a)\}$ are complex structures on $X$, with $\mathcal{A}$ being the one I proved to exist, which comes straight out of the definition of local homeomorphism. Moreover I write $\mathcal{B}=\{(V_{\alpha},\psi_{\alpha})\}$ for the complex structure on $Y$.
I need to prove that the transition functions from $\mathcal{A}$ to $\mathcal{C}$ are homeomorphic. For $x\in W_a$ I consider $U_x\cap W_a$. It clearly suffices to prove that $\chi_a\phi_x^{-1}$ and $\phi_x\chi_a^{-1}$ are holomorphic on the appropriate images of $U_x\cap W_a$.
Now $f|_{U_x\cap W_a}$ is a homemorphism, so in particular I know that $\psi_{\alpha}f\chi_a^{-1}$ and $\psi_{\alpha}f\phi_x^{-1}$ are holomorphic and invertible. I now need only to prove that the inverses of these are themselves holomorphic and I'm done.
I can't see how to do this however! Has anyone got any good ideas? Many thanks!