Suppose we have two distinct $\gamma_1$ and $\gamma_2$, non-intersecting geodesic lines with disjoint endpoints on the hyperbolic plane. Assuming $\gamma_i$ cuts the hyperbolic plane into two half-planes $A_i$, $B_i$ satisfying $A_1 \subset A_2$ and $B_2 \subset B_1$.
On the disc model this just looks like the disc with two geodesics which don't intersect, with endpoints on the boundary, I think.
There is a Mobius map $f$ with $f(B_1) = B_2$
How do we show this map exists and it is necessarily hyperbolic?
So far, I have argued we can send the endpoints of one geodesic to the endpoints of the other in a nice way (so they don't flip) and since Mobius maps preserve orientation, we get $f(B_1) = B_2$ but I don't know if this is sufficient.
As for showing map is hyperbolic, I don't know where to begin. I think we have to use the fact that a map from the interval to itself has a fixed point