Free coactions of a finite dimensional $C^{\ast }$-Hopf algebra and strong Morita equivalence

We shall introduce a notion of free coactions of a finite dimensional $C^{\ast }$-Hopf algebra on a $C^{\ast }$-algebra modifying a notion of free actions of a discrete group on a $C^{\ast }$-algebra and we shall study several properties on coactions of a finite dimensional $C^{\ast }$-Hopf algebra on $C^{\ast }$-algebras, which are relating to strong Morita equivalence for inclusions of $C^{\ast }$-algebras.