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We study the structure of C*-algebras associated with compactly alignedproduct systems over group embeddable right LCM-semigroups. Towards this end weemploy controlled maps and a controlled elimination method that associates theoriginal cores to those of the controlling pair, and we combine withapplications of the C*-envelope theory for cosystems of nonselfadjoint operatoralgebras recently produced. We derive several applications of these methodsthat generalize results on single C*-correspondences.  First we show that if the controlling group is exact then the co-universalC*-algebra of the product system coincides with the quotient of the FockC*-algebra by the ideal of strong covariance relations. We show that if thecontrolling group is amenable then the product system is amenable. Inparticular if the controlling group is abelian then the co-universal C*-algebrais the C*-envelope of the tensor algebra.  Secondly we give necessary and sufficient conditions for the Fock C*-algebrato be nuclear and exact. When the controlling group is amenable we completelycharacterize nuclearity and exactness of any equivariant injectiveNica-covariant representation of the product system.  Thirdly we consider controlled maps that enjoy a saturation property. In thiscase we induce a compactly aligned product system over the controlling pairthat shares the same Fock representation, and preserves injectivity. By usingco-universality, we show that they share the same reduced covariance algebras.If in addition the controlling pair is a total order then the fixed pointalgebra of the controlling group induces a super-product system that has thesame reduced covariance algebra and is moreover reversible.

Co-universality and controlled maps on product systems over right LCM-semigroups