be preserved by tensoring on either side as well as by the various fusion morphisms. So for simplicity we suppose that it is preserved by all functors; equivalently, that it is a split epimorphism. (When the fusion morphism is unital as well as counital, and so corresponds to a bimonoid, this is equivalent to the usual unitality condition for an action.) In Section \[sec:tric\_module\] we investigate additional assumptions on a counital fusion morphism, weaker than unitality, under which such modules constitute a monoidal category admitting a strict monoidal forgetful functor to the base category. Again, modules over a bimonoid (i.e. over a unital and counital fusion morphism) constitute a monoidal category because they induce [*bimonads*]{} (that is, opmonoidal monads). This is no longer true for a counital but non-unital fusion morphism. We describe a generalization of the notion of opmonoidal monad which can be used to explain the monoidal structure in this case.

Assuming [*regularity*]{} of a multiplier bimonoid in a braided monoidal category, its comodules and modules are defined in the respective Sections \[sec:comodule\] and \[sec:module\] as objects carrying compatible (co)module structures over appropriate pairs of the fusion morphisms $t_1, t_2, t_3$ and $t_4$. They are shown to constitute 