Non-commutative ambits and equivariant compactifications

We prove that an action $\rho:A\to M(C\_0(\mathbb{G})\otimes A)$ 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 $\mathbb{G}$-equivariant compactifications: that the categories compactifications of $\rho$ 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 $\mathbb{G}$ is regular, coamenable we also show that the forgetful functor from unital $\mathbb{G}$-$C^$-algebras to unital $C^$-algebras creates finite limits and is comonadic and that the monomorphisms in the former category are injective.