While reading Donaldson's book on Riemann surfaces, I came across the following proposition.
Proposition
Let $H$ be upper half plane, $p,\tilde{p} \in H$, and $f: N \rightarrow \tilde{N}$ be a diffeomorphism from an open neighborhood of $p$ to a neighborhood of $\tilde{p}$. Then, if $f$ is an oriented isometry of the hyperbolic metric, it is the restriction of a map defined by an element of $PSL(2,\mathbb{R})$ .
I tried to prove it, but could not. Could you please tell me how to show it? Thank you very much.