  Given a horizontal monoid M in a duoidal category F, we examine the
relationship between bimonoid structures on M and monoidal structures on the
category of right M-modules which lift the vertical monoidal structure of F. We
obtain our result using a variant of the Tannaka adjunction. The approach taken
utilizes hom-enriched categories rather than categories on which a monoidal
category acts ("actegories"). The requirement of enrichment in F itself demands
the existence of some internal homs, leading to the consideration of
convolution for duoidal categories. Proving that certain hom-functors are
monoidal, and so take monoids to monoids, unifies classical convolution in
algebra and Day convolution for categories. Hopf bimonoids are defined leading
to a lifting of closed structures. Warped monoidal structures permit the
construction of new duoidal categories.
