  Let X be a product system over a quasi-lattice ordered group. Under mild
hypotheses, we associate to X a C*-algebra which is co-universal for injective
Nica covariant Toeplitz representations of X which preserve the gauge coaction.
Under appropriate amenability criteria, this co-universal C*-algebra coincides
with the Cuntz-Nica-Pimsner algebra introduced by Sims and Yeend. We prove two
key uniqueness theorems, and indicate how to use our theorems to realise a
number of reduced crossed products as instances of our co-universal algebras.
In each case, it is an easy corollary that the Cuntz-Nica-Pimsner algebra is
isomorphic to the corresponding full crossed product.
