If $(H,\Delta,\nabla)$ is a Hopf algebra in the prebraided monoidal category $(\mathcal{C},\Psi)$ then
$\Psi_{H,H}=\left(\nabla\otimes \nabla\right)\left(S\otimes\Delta \nabla\otimes S\right)\left(\Delta\otimes\Delta\right)$
Is it true that
$\Psi^{-1}_{H,H}=\left(\nabla\otimes \nabla\right)\left(S^{-1}\otimes\Delta \nabla\otimes S^{-1}\right)\left(\Delta\otimes\Delta\right)$
?
(We don't assume that the category is symmetric).
If it is true, is there any simple method to prove it ? I tried to use diagrammatic methods, but in my diagrams was too many lines, and I had a problem to see anything.
This is not true in general. Instead, the equation $$ \Psi^{-1}=(\nabla\Psi^{-1}\otimes \nabla\Psi^{-1})(S^{-1}\otimes \Delta\nabla\Psi^{-1}\otimes S^{-1})(\Delta \otimes \Delta) $$ holds. This may not be very useful as $\Psi^{-1}$ appears on either side. Note that $S^{-1}$ only satisfies the twisted convolution invertibility $$ \nabla\Psi^{-1}(S^{-1}\otimes \operatorname{id})\Delta=\nabla\Psi^{-1}(\operatorname{id}\otimes S^{-1})\Delta=1\varepsilon. $$ To prove such equations effectively, one can use a graphical calculus as for example in [Majid: Foundations of Quantum Group Theory] from 9.2 onwards. I attach the computation as a photo (sorry about poor quality).