Symmetric self-adjoint Hopf categories and a categorical Heisenberg double

We define what we call a symmetric self-adjoint Hopf structure on a semisimple abelian category, which is an analog of Zelevinsky's positive self-adjoint Hopf algebra structure for categories. As examples we exhibit this structure on the categories of polynomial functors and equivariant polynomial functors and obtain a categorical manifestation of Zelevinsky's decomposition theorem involving them. It follows from the work of Zelevinsky that every positive self-adjoint  Hopf algebra A  admits a Fock space action of the Heisenberg double (A,A).. We show that the notion of symmetric self-adjoint Hopf category leads naturally to the definition of a categorical analog of such an action and that every symmetric self-adjoint Hopf category admits such an action.