This appears in the section 3.7 of the book Compact Riemann Surfaces by Jurgen Jost, right after Lemma 3.7.3. The exact words are
Now let $v:\Sigma_1\to\Sigma_2$ be a Lipschitz continuous map. Cover $\Sigma_1$ by coordinate neighborhoods. Choose $R_0<1$ so that, for every $z_0\in\Sigma_1$, a disc of the form $$B(z_0,R_0)=\{z:|z-z_0|<R_0\}$$ is contained inside a coordinate neighborhood.
Note that the only mention of $\Sigma_1$ before the cited paragraph assumes $\Sigma_1$ to be a compact surfaces, and judging from how these arguments are to be used later, $\Sigma_1$ is undoubtedly a compact Riemann surface (with no given metric).
My questions are:
(1) What is a Lipschitz continuous map from a surface in this context?
(2) What is the absolute value used in the representation of $B(z_0,R_0)$?
(3) It seem that the author is trying to identify (locally) $\Sigma_1$ with its local coordinates. However, if we do so, two problems arise: first, since we can multiply a chart by a constant, we can modify the distance of two points (covered by the same chart, distance given by the absolute value) to any distance we want, and the Lipschitz bound is not well-defined; second, for the same reason above, there's no need to define $R_0$.
I have searched this section over and again for what the absolute value on $\Sigma_1$ is but I can't come up with a plausible explanation.
Any help is greatly appreciated.