  A cosystem consists of a possibly nonselfadoint operator algebra equipped
with a coaction by a discrete group. We introduce the concept of C*-envelope
for a cosystem; roughly speaking, this is the smallest C*-algebraic cosystem
that contains an equivariant completely isometric copy of the original one. We
show that the C*-envelope for a cosystem always exists and we explain how it
relates to the usual C*-envelope. We then show that for compactly aligned
product systems over group-embeddable right LCM semigroups, the C*-envelope is
co-universal, in the sense of Carlsen, Larsen, Sims and Vittadello, for the
Fock tensor algebra equipped with its natural coaction. This yields the
existence of a co-universal C*-algebra, generalizing previous results of
Carlsen, Larsen, Sims and Vittadello, and of Dor-On and Katsoulis. We also
realize the C*-envelope of the tensor algebra as the reduced cross sectional
algebra of a Fell bundle introduced by Sehnem, which, under a mild assumption
of normality, we then identify to the quotient of the Fock algebra by the image
of Sehnem's strong covariance ideal. In another application, we obtain a
reduced Hao-Ng isomorphism theorem for the co-universal algebras.
