In this note we look as diffeomorphisms confining ourselves for now to inverse functions. There is a local diffeomorphism where both first partial derivatives of f(w,z) are non-zero so both dw/dz and dz/dw exist and are unique. For there to be a diffeomorphism between regions it is necessary that these conditions hold throughout. We have seen how to get such regions by joining nodes together to form boundaries for them. It may be useful to recall here that in general only some boundaries imaging cuts run through  nodes and serve to separate ranges of branches. These may be called proper boundaries.  Those that do not, terminating in ordinary points or internal poles, may be called internal boundaries or self-boundaries.
This was done for the w-plane and in general the same thing needs to be done for the z-plane.
However for inverse functions nothing more actually needs to be done because here the z-plane does not contain any nodes.Let f(w,z) be irreducible and written in the form zp(w)+q(w). Then the partial derivative with respect to z is p(w) and if this has a zero then q(w) has the same zero contrary to the assumption of irreducibility. Now restrict w to the range of any one branch. Then the mapping is 1 to 1, dz/dw can be calculated from the equation f(w,z)=0 and is never either infinite or zero so the mapping is a diffeomorphism. This applies for each range. Also there is a diffeomorphism between the range of any branch and any other which is mediated by their mappings to the cut z-plane.