Forster gives a proof of the local behavior of holomorphic mappings as the following. Let $f: X\to Y$ be a nonconstant holomorphic mapping where $X, Y$ are Riemann surfaces. Suppose $a\in X$ and $b=f(a)$, then there exist charts $(U, \varphi)$ on $X$ and $(U',\varphi')$ on $Y$ such that
(1) $a\in U, \varphi(a)=0$, $b\in U',\varphi'(b)=0$;
(2) $f(U)\in U'$;
(3) The map $F=\varphi'\circ f\circ\varphi^{-1}:V\to V'$ is given by $f(z)=z^k$ for all $z\in V$.
First, he notes that there exist charts $\varphi_1: U_1\to V_1$ and $\varphi': U'\to V'$ that suffice the first two properties if we replace $(U,\varphi)$ by $(U_1, \varphi_1)$. Then he claims that $f_1=\varphi'\circ f\circ\varphi^{-1}$ is nonconstant by identity theorem. Since $f_1(0)=0$, there exists some $k$ such that $f(z)=z^kg(z)$ for $z\in V\ni 0$, where $g(0)\neq 0$ is holomorphic. Then there is some holomorphic $h$ such that $h^k=g$...
I could understand the part after this construction. There are three steps that are not that clear to me. First, the existence of $\varphi_1:U_1\to V_1$. Second, in what way the identity theorem is applied to our case so that we can conclude $f_1$ is nonconstant (my guess is that if $f_1$ is constant then $\varphi^{-1}\circ f_1\circ\varphi$ will be constant and since this composition coincides with $f$ on the open set $U$, $f$ will then be constant by identity theorem, a contradiction)? Third, where does the function $h$ come from?
I tried to read some other sources, but the proof here seems to be pretty standard. Thanks in advance. Any help will be appreciated.