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 $\beta_{A}$?