\draw[braid] (s3t) to[out=270,in=45] (t1d);
\draw[braid] (s2t) to[out=270,in=45] (t1u); 
\draw[braid] (s1t) to[out=315,in=135] (t1u);
\draw[braid] (s0t) to[out=270,in=180] (m) to[out=0,in=225] (t1u);
\draw[braid] (m) to[out=270,in=90] (s1b);
\draw[braid] (t1u) to[out=315,in=135] (t1d);
\draw[braid] (t1d) to[out=250,in=100]  (s2b);
\draw[braid] (t1d) to[out=290,in=80] (s3b);
\end{tikzpicture}$$ from which we conclude by the non-degeneracy of $m$ that the fusion equation $(t_1.1)t_1^{13}(1.t_1)=(1.t_1)(t_1.1)$ holds. Dually, $t_2$ is a counital fusion morphism in $\vec\rev$.

\[prop:multiplication\] For a counital fusion morphism $(t:A^2\to A^2,e:A\to I)$ in a braided monoidal category $\C$, the following assertions hold.

-   There is an associative multiplication $
    m:=\big(
    \xymatrix@C=20pt{
    A^2\ar[r]^-t&
    A^2\ar[r]^-{e.1}&
    A}\big).
    $

-   $(1.m)(t.1)=t(1.m)$.

-   $(m.1)(b^{-1}.1)(1.t)(b.1)(1.t)=t(m.1)$.

-   $em=e.e$.

In the following diagram, the top region commutes by the fusion equation and the triangular region commutes by the counitality condition. The bottom left region commutes by functoriality of the monoidal product and coherence of the braiding. $$\xymatrix{
A^3\ar[rrrr]^-{t.1}\ar[d]_-{1.t}&&&&
A^3
\ar[d]^-{1.t}&
\\
A^3\ar[r]^-{b.1}\ar[d]_-{1.e.1}&
A^3\ar[r]^-{1.t}&
A^3\ar[r]^-{b^{-1}.1}&
A^3\ar[r]^-{t.1}\ar[rd]_-{1.e.1}&
A^3
\ar[d]^-{1.e.1}&
\\
A^2\ar[rrrr]_-t&&&&
A^2}$$ This proves part (2) and postcomposing both paths by $e.1$ we obtain part (1) by functoriality of the monoidal product. In order to prove (3), postcompose both sides of the fusion equation by $e.1.1$ and use functoriality of the monoidal product. For (4), postcompose both sides of the counitality condition by $e$ 