  We introduce the theory IH of interacting Hopf algebras, parametrised over a
principal ideal domain R. The axioms of IH are derived using Lack's approach to
composing PROPs: they feature two Hopf algebra and two Frobenius algebra
structures on four different monoid-comonoid pairs. This construction is
instrumental in showing that IH is isomorphic to the PROP of linear relations
(i.e. subspaces) over the field of fractions of R.
