translate to commutativity of the respective diagrams $$\label{eq:intro_comod}
\xymatrix{
V.A.A \ar[r]^-{1.t_1}\ar[d]_-{v.1}&
V.A.A \ar[r]^-{b.1}&
A.V.A \ar[r]^-{1.v}&
A.V.A \ar[r]^-{b^{-1}.1}&
V.A.A\ar[d]^-{v.1}&
V.A\ar[r]^-v\ar[rd]_-{1.e}&
V.A\ar[d]^-{1.e}\\
V.A.A \ar[rrrr]_-{1.t_1}&&&&
V.A.A&&
V}$$ (where $e$ denotes the counit of $A$). By duality, there is a symmetric description, in terms of $t_1$, of left modules over the monoid $A$.

A key example of a bialgebra (in the symmetric monoidal category $\vec$ of vector spaces over a field $k$) is given by the algebra $k^G$ of all $k$-valued functions on a finite monoid $G$. In the case of functions of finite support on an infinite monoid $G$, we still have the multiplication and the counit of $k^G$, but not the unit or comultiplication; instead of a bialgebra, under a mild cancellation assumption we have a [*multiplier bialgebra*]{}. Motivated by this, we consider various weakenings of the notion of bimonoid in a braided monoidal category.

First we ask what happens to the above picture if we give up self-duality and consider a counital, but no longer unital fusion morphism $t_1:A.A\to A.A$? By this we mean that $t_1$ satisfies the fusion equation and is equipped with a morphism (the [*counit*]{}) $e$ from $A$ to the monoidal unit; such that $(1.e)t_1=1.e$. We do not 