Abstract
We use purity, a principle borrowed from the foundations of quantum information, to show that all isometric comonoids in the category CPM(fHilb) are necessarily pure. As a corollary, we answer an open question about special dagger Frobenius algebras (and classical structures in particular) in CPM(fHilb): we show that they are all canonical, i.e. that they all arise by doubling of special dagger Frobenius algebras from the category fHilb.