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