Let $(H, \nu,\eta, \Delta, \epsilon, S)$ be a Hopf algebra. $S$ is the antipode.
I am reading a proof of the fact $S(xy)=S(y)S(x)$.
First, define maps $\nu, \rho$ in $\hom(H \otimes H, H)$ by
$\nu(x \otimes y)=S(y)S(x)$ and $\rho(x \otimes y)=S(xy)$.
To prove the fact, we have to show that $\rho=\nu$. It is written that to show this it suffices to show that $\rho* \mu=\mu*\nu=\eta\epsilon$, where $*$ is the convolution.
I don't understand why this implies that $\rho=\nu$.
Please give some advice.