  Building on the recent computation of the cohomology rings of smooth toric
varieties and partial quotients of moment-angle complexes, we investigate the
naturality properties of the resulting isomorphism between the cohomology of
such a space and the torsion product involving the Stanley-Reisner ring. If 2
is invertible in the chosen coefficient ring, then the isomorphism is natural
with respect to toric morphisms, which for partial quotients are defined in
analogy with toric varieties. In general there are deformation terms that we
describe explicitly.
