Let $(H, \mu, \eta, \Delta, \varepsilon, S)$ be a Hopf algebra. Let $\mathcal F \in H \otimes H$ be an invertible element such that
$(1)$ $(\mathcal F \otimes 1) (\Delta \otimes \text {id}) (\mathcal F) = (1 \otimes \mathcal F) (\text {id} \otimes \Delta) (\mathcal F),$
$(2)$ $(\varepsilon \otimes \text {id}) (\mathcal F) = 1 = (\text {id} \otimes \varepsilon) (\mathcal F).$
Then the element $v = \mu (\text {id} \otimes S) (\mathcal F) \in H$ is invertible with inverse being given by $\mu (S \otimes \text {id}) (\mathcal F^{-1}).$
My Attempt $:$
Since $\Delta$ and $\varepsilon$ are algebra homomorphisms it follows that
$(\Delta \otimes \text {id}) (\mathcal F^{-1}) (\mathcal F^{-1} \otimes 1) = (\text {id} \otimes \Delta) (\mathcal F^{-1}) (1 \otimes \mathcal F^{-1}),$
and,
$(\varepsilon \otimes \text {id}) (\mathcal F^{-1}) = 1 = (\text {id} \otimes \varepsilon) (\mathcal F^{-1}).$
Using these relations and the axioms of Hopf algebra I managed to show that $v u = 1$ where $u = \mu (S \otimes \text {id}) (\mathcal F^{-1}).$ But I am having hard time showing that $u v = 1.$
Any help in this regard would be greatly appreciated. Thanks a bunch!
Source $:$ A Guide to Quantum Groups by Vyjayanthi Chari and Andrew Pressley.