Here is a exercise in the Eiichi Abe's book Hopf Algebras, Ex2.4,Ch2, p.83.
If $X$ is a left coideal, show that $X\wedge Y$ is a right coideal and that $X\subset X\wedge Y$.
Maybe you can explain the $X\wedge Y$ in the Hopf algebra, does it has any geometry meaning?