  Motivated by the work of of A. Zelevinsky on positive self-adjoint Hopf
algebras, we define what we call a symmetric self-adjoint Hopf structure for a
certain kind of semisimple abelian categories. It is known that every positive
self-adjoint Hopf algebra admits a natural action of the associated Heisenberg
double. We construct canonical morphisms lifting the relations that define this
action on the algebra level and define an object that we call a categorical
Heisenberg double that is a natural setting for considering these morphisms. As
examples, we exhibit the symmetric self-adjoint Hopf structure on the
categories of polynomial functors and equivariant polynomial functors. In the
case of the category of polynomial functors we obtain categorification of the
Fock space representation of the infinite-dimensional Heisenberg algebra.
