Find conformal map $h$ from the circular sector common to circles $ | z-1| = 1$ and $ |z-i| = 1$ to the right-half plane. Find $h^{-1}(1)$

Question

I am trying to first construct a fractional linear transformation, to map the sector to a half or full circle. From there it seems straightforward using rotations, translations, and standard mappings between circles to half-planes to create a map that takes the sector to the right-half plane.

My question is, how do I construct the fractional linear transformation? Lang's Complex Analysis shows how to construction fractional linear transformations to map a triplet of points to another triplet of points, but I don't see how that would be useful here.