Suppose that $H$ is a finite dimensional Hopf algebra (with counit $\varepsilon$) and $T$ is a non zero right integral of $H^{\star}$ (the dual Hopf algebra). Let $ad_h$ be the adjoint action on $H$, that is, $ad_h(k)=h_1kS(h_2)$, where $S$ is the antipode map.

When $T\circ ad_h=\varepsilon(h)T$, for all $h\in H$? Is it holds when $H$ is cosemisimple?

For example, if $H=kG$ is the group algebra, then the equation holds for all $h\in H$. Indeed, in this case, the set of integrals (left or right) is generated by $T(g)=\delta_{g,1_G}$ and is immediate to check the equation.