  We establish necessary and sufficient conditions for the realization of
mapping schemata as post-critically finite polynomials, or more generally, as
post-critically finite polynomial maps from a finite union of copies of the
complex numbers {\bf C} to itself which have degree two or more in each copy.
As a consequence of these results we prove a transitivity relation between
hyperbolic components in parameter space which was conjectured by Milnor.
