)$ is normal precisely when the following is satisfied: a surjective homomorphism $\pi:{\operatorname{\mathcal{NO}}_{\!X}}\to B$ is injective if and only if $\pi$ is injective on ${\operatorname{\mathcal{NO}}_{\!X}}^\nu$ and there is a coaction $\beta$ on $B$ such that $\pi$ is $\nu-\beta$ equivariant.  Apply Corollary \[cor:characterise-normal\].  