Antipode

Remarks

The convolution of linear maps $f, g \colon H \to H$ is $f * g = m \circ (f \otimes g) \circ \Delta$. It makes $\End_K(H)$ an associative algebra with identity $\eta \circ \varepsilon$, and the condition says $S * \id_H = \eta \circ \varepsilon = \id_H * S$. So $S$ is a two-sided inverse of $\id_H$, and an antipode is unique when it exists. Example: $H = K[x]$ with $\Delta(x) = x \otimes 1 + 1 \otimes x$ and $\varepsilon(x) = 0$ has the antipode $S(x^n) = (-x)^n$. On $x$ the check is $S(x) \cdot 1 + S(1) \cdot x = -x + x = 0 = \varepsilon(x) 1$. Non-example: the identity map of $K[x]$ is not an antipode, since $m(\id \otimes \id)\Delta(x) = 2x \ne 0$ when the characteristic of $K$ is not $2$. An antipode is an algebra anti-homomorphism, $S(ab) = S(b)S(a)$, and a coalgebra anti-homomorphism. If $H$ is commutative or cocommutative, then $S \circ S = \id_H$.