For the monoidal product of several copies of the same object also the power notation is used: $A.A=A^2$. Composition of morphisms is denoted by juxtaposition and the identity morphism (at any object) is denoted by $1$. We do not assume that the monoidal structure is strict but — relying on coherence — usually we omit explicitly denoting the associativity and unit isomorphisms. For a braided monoidal category $\C$, we denote by $\overline \C$ the same monoidal category $\C$ with the inverse braiding $(b_{Y,X})^{-1}:X.Y\to Y.X$. The [*reverse*]{} $\C\rev$ of $\C$ means the same category $\C$ with the opposite monoidal product $(X,Y) \mapsto Y.X$ (thus the same monoidal unit $I$) and the braiding $b_{Y,X}:Y.X\to X.Y$. The braided monoidal categories $\overline \C\rev$ and $\overline {\C\rev}$ clearly coincide.

Counital fusion morphisms {#sec:tricoc}
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Motivated by our definition in Section \[sec:multiplier\_bimonoid\] of multiplier bimonoid in a braided monoidal category, in this section we study counital fusion morphisms possibly without unit. To any such creature we associate a monoidal category of appropriately defined comodules. Under further assumptions — about certain morphisms being epimorphic — we associate to it a second monoidal category of suitable modules. The monoidality of these categories is explained by the structure of 