We tenuously connect three results:
1) Jeff Egger has developed a construction of functor categories between star-autonomous categories which specialises to cover linearly-distributive categories in some cases.
2) Craig Pastro and Ross Street have shown that, for F a separable frobenius monoid in a braided monoidal category, there is a weak bialgebra structure on the tensor product of F with itself.
3) In my thesis, I show that every separable Frobenius monoidal functor gives rise to a weak-bialgebra in its codomain.
Modulo some technical wrinkles which we will discuss, it appears as though the Tannaka construction of 3) can be obtained as a composite of processes 1) and 2).