# Braided Hopf algebra - properties of braiding

Let $(H,\nabla,\Delta,S)$ be a Hopf algebra in a braided category.

I'm trying to simplify the following

$(\nabla\otimes \mathrm{id}\otimes \mathrm{id})(\nabla \otimes \Psi \otimes \nabla)(\mathrm{id}\otimes \Psi \otimes \Psi \otimes \mathrm{id})(\Delta\otimes \Delta \otimes \Delta)$

and eliminate a braiding from this relation.

I used relation between $\Psi$ and $S$, i.e. $\Psi=\left(\nabla\otimes \nabla\right)\left(S\otimes\Delta \nabla\otimes S\right)\left(\Delta\otimes\Delta\right)$ and obtain big diagram, which I can't simplify. I've a problem with eliminate $S$'s which are in the center of the graph.

Is there any simply method ?

(It seems to me that it should remain only one $S$, but I don't see how to obtain it.)

EDIT: I did it last night using 14 pages of paper and very large diagrams :) If someone sees a simplest method, please give me a hint.

You can use associativity, and the the bialgebra condition to get $$(\nabla\otimes \operatorname{id}\otimes \operatorname{id})(\operatorname{id}\otimes \Psi\otimes \operatorname{id})(\Delta\otimes \Delta\nabla),$$ and then your identity to eliminate the braiding only once.