[@BoGTLC:wmba Theorem 2.11]. Based on [@Bohm:wmba_comod Theorem 1.2] and [@BoGTLC:wmba Proposition 2.6], it can be described as follows. A multiplier bialgebra is given by a vector space $A$ equipped with an associative but not necessarily unital multiplication $m:A^2\to A$; which is required to be surjective and non-degenerate in the sense that both maps $A\to \mathsf{End(A)}$, $a\mapsto
m(a.-)$ and $a\mapsto m(-.a)$ are injective. Furthermore, the existence of linear maps $t_1,t_2:A^2\to A^2$ and $e:A\to k$ is required such that the following axioms hold (where $b$ denotes the symmetry in $\mathsf{vec}$; of course its components satisfy $b^2=1$).

-   $t_1(m.1)=(m.1)(b.1)(1.t_1)(b.1)(1.t_1)$; equivalently,\
    $t_2(1.m)=(1.m)(1.b)(t_2.1)(1.b)(t_2.1)$.

-   Both maps $(m.1)(b.1)(1.t_1)$ and $(1.m)(1.b)(t_2.1)$ are surjective.

-   $em=e.e$.

-   $(t_2.1)(1.t_1)=(1.t_1)(t_2.1)$.

-   $(e.1)t_1=m=(1.e)t_2$.

We claim that $t_1$ is then a counital fusion morphism in $\mathsf{vec}$. Indeed, by (e) and (d), $$\label{eq:m_t_1-2}
(m.1)(1.t_1)=(1.m)(t_2.1).$$ Postcomposing with $1.e$, we obtain by (c) and (e) $$m[1.(1.e)t_1]=
(1.e)t_2.e=
m(1.1.e),$$ from which we conclude by the non-degeneracy of $m$ that the counitality condition $(1.e)t_1=1.e$ holds. Furthermore, $$\label{eq:short_fusion}
\begin{tikzpicture}
    \path (1,2) node[arr,name=t1u] {$t_1$} 
(1,1) node[arr,name=t1d]  {$t_1$} 
(0,0.3) node[arr,name=t1l] {$t_1$} 
(-0.5,-0.2) node[empty,name=m] {};
\path (1.1,2.7) node[empty,name=s3t] {} 
(0.5,2.7) node[empty,name=s2t] {} 
(0,2.7) node[empty,name=s1t] {} 
(-0.5,-0.5) node[empty,name=s1b] {} 
(0.5,-0.5) node[empty,name=s2b] {} 
(1,-0.5) node[empty,name=s3b] {} 
(-0.5,2.7) node[empty,name=s0t] {};
\draw[braid] (s3t) to[out=315,in=45] (t1u);
