qa.quantum algebra - Coaction on the Universal Calculus

Let $A$ be an algebra, $H$ a Hopf algebra, and
$$
beta_A: A to A otimes H, ~~~~~ a mapsto a^{(1)} otimes a^{(2)}
$$
a right $H$-coaction. This induces a right $H$-coaction on $A otimes A$ defined by
$$
beta_{A otimes A}: a otimes b mapsto a^{(1)} otimes b^{(1)} otimes a^{(2)}b^{(2)}.
$$
My question is: Does this restrict to a coaction on the universal calculus over $A$, namely to a $H$-coaction on the kernel of the multiplication map $m:A otimes A to A$? I feel this is a very simple question but I can't seem to find an answer.

If the construction does not work, does anyone know of a way to induce a coaction on the universal calculus over $A$ from $beta_{A}$?