notions of module and comodule. The unit of the monoid $A$ does not occur in the diagrams in . Hence they can be used to define comodules also for only counital fusion morphisms. We study this situation in Section \[sec:tric\_comod\], where we show that such comodules constitute a monoidal category admitting a strict monoidal forgetful functor to the base category. Comodules over a bimonoid (i.e. over a unital and counital fusion morphism) constitute a monoidal category because they induce [*bicomonads*]{} (that is, monoidal comonads). This is no longer true for a counital but non-unital fusion morphism. We describe a generalization of the notion of monoidal comonad which can be used to explain the monoidal structure in this case.

It is somewhat more delicate what should be a module over a counital but non-unital fusion morphism. Replacing an associative (left) action, we require the existence of a morphism $q:A.Q\to A.Q$ satisfying an appropriate fusion type equation. But if $A$ has no unit, it is not immediate what should replace the unitality of an action. One possibility is to require that the morphism $$\xymatrix{
A.Q\ar[r]^-{q}&
A.Q\ar[r]^-{e.1}&
Q}$$ be an epimorphism. But in order to develop the theory, we need this epimorphism to 