equality we used naturality of the braiding, associativity of the multiplication, and coassociativity of the comultiplication. In the second equality we used multiplicativity of the comultiplication. The counitality condition follows by the calculation

$$\begin{tikzpicture}[every node/.style={empty}]     \path (0.25,1.5) node[name=s1t] {}
(1.0,1.5) node[name=s2t] {}
 (0,0) node[name=sb] {} 
(0.25,1.0) node[name=d] {} 
(0.5,0.75) node[name=v] {}
(0.75,0.5) node[name=m] {};
\draw (0.75,0.2) node[name=e] {$\circ$};
\draw[braid] (s2t) to[out=270,in=0] (m) to[out=180,in=270] (v) to[out=90,in=0] (d) to[out=180,in=90] (sb);
\draw[braid] (s1t) to (d) ;
\draw (e) to (m);
\draw (1.5,0.75) node{$=$};
\draw (1.75,0.75) node{};
\end{tikzpicture}
\begin{tikzpicture}[every node/.style={empty}]     \path (0.25,1.5) node[name=s1t] {}
(0.75,1.5) node[name=s2t] {}
 (0,0) node[name=sb] {} 
(0.25,1.0) node[name=d] {} ;
\draw (0.5,0.2) node[name=e] {$\circ$};
\draw (0.75,0.2) node[name=e2] {$\circ$};
\draw[braid] (e) to[out=90,in=0] (d) to[out=180,in=90]  (sb);
\draw[braid] (s1t) to (d) ;
\draw (e2) to (s2t);
\draw (1.25,0.75) node{$=$};
\draw (1.75,0.75) node{};
\end{tikzpicture}
\begin{tikzpicture}[every node/.style={empty}]     \path (0.25,1.5) node[name=s1t] {}
(0.75,1.5) node[name=s2t] {}
 (0.25,0) node[name=sb] {} ;
\draw (0.75,0.2) node[name=e] {$\circ$};
\draw[braid] (s1t) to (sb) ;
\draw (e) to (s2t);
\end{tikzpicture}$$

where the first equality uses multiplicativity of the counit and the second one counitality of the comultiplication.

Further examples, no longer unital, come from multiplier bialgebras over a field:

\[ex:multiplier\_bialg\] By a [*multiplier bialgebra*]{} over a field $k$ we mean the structure in 