Proving that norm is holomorphic for compact riemann surfaces.

Question

Let $h : X\rightarrow Y$ be a map of compact riemann surfaces. We define $norm_h: \mathbb C (X ) \rightarrow \mathbb C (Y)$ as $norm_h(f)(y)=\prod _{x\in h^{-1} (y)} f(x)^{e_x}$, here $e_x$ is the ramification degree. How do I see that this map is meromorphic?
I understand this if we restrict to the case where there are no exceptional (ramification $>1$) points.