Non-commutative ambits and equivariant compactifications

We prove that an action ρ W A ! M.C₀.G/ ˝ A/ of a locally compact quantum group on a C ^∗-algebra has a universal equivariant compactification and prove a number of other category-theoretic results on G-equivariant compactifications: that the categories compactifications of ρ and A, respectively, are locally presentable (hence complete and cocomplete), that the forgetful functor between them is a colimit-creating left adjoint, and that epimorphisms therein are surjective and injections are regular monomorphisms. When G is regular, coamenable we also show that the forgetful functor from unital G-C ^∗- algebras to unital C ^∗-algebras creates finite limits and is comonadic and that the monomorphisms in the former category are injective.