A question on a Hopf algebra

Question

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.

Answer 1

In an associative algebra, if an element has a left inverse and a right inverse, these two are equal.

Now $\rho\star\mu=\eta\epsilon$ and $\mu\star\nu=\eta\epsilon$ mean precisely that $\rho$ and $\nu$ are left and right inverse to $\mu$, respectively, in the convolution algebra $\hom(H\otimes H,H)$ (recall that $\eta\epsilon$ is the unit element in that algebra)