Holomorphic map between Riemann surfaces

Summary

A holomorphic map between Riemann surfaces is a continuous map that is holomorphic when written in local coordinates on both sides. These maps are the morphisms of the category of Riemann surfaces; the bijective ones are the biholomorphisms, which identify two Riemann surfaces. Meromorphic functions on a Riemann surface are the same as holomorphic maps to the Riemann sphere that are not constantly $\infty$.