Let $U\subset X$ be an open subset of a connected Riemann surface $X$. Let $z:U\longrightarrow B(0,1)$ be a diffeomorphism, where $B(0,1)$ is the open unit disc in $\mathbf{C}$. Let $P\in U$ be the unique point such that $z(P) =0$.
Suppose I take another point $Q\in U$. I want to use $z:U\longrightarrow B(0,1)$ to construct a coordinate around $Q$.
Question. Does the following work?
Consider an open set $V$ in $U$ whose image under $z$ is a small open disc around $z(Q)$. Then we define $w: V\longrightarrow B(0,1)$ by $w(x) = z(x) - z(Q).$ Is this is a coordinate around $Q$?