A holomorphic function $\phi:X\to Y$, where $X$ and $Y$ are Riemann surfaces, is described in the following way:     For $a\in X$, let $a\in U_1$, where $U_1$ is an open set. Let $C_1:U_1\to V_1$ be a chart. Similarly, let $f(a)\in U_2$, and let $C_2:U_2\to V_2$ be a chart. Then $\phi$ is holomorphic iff $(C_2\circ \phi\circ C_1^{-1}): V_1\to V_2$ is holomorphic.    Another definition of holomorphic functions that is given is the following:     For any holomorphic function $f$ on $Y$, if $f\circ \phi$ is also holomorphic on $X$, then $\phi$ is holomorphic.    How are the two definitions equivalent?
I'm trying to show the following:  Let $f:X\to Y$ be a proper holomorphic map between connected, non-empty Riemann Surfaces. Show that a map $g:Y\to\mathbb{C}$ is  holomorphic if and only if  its composition with f is holomorphic.  So far I know that $f$ is surjective, since it's proper and holomorphic. My next step was to look at the charts, but it's there, I get stuck.
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