Hopf Quasigroup Galois Extensions and a Morita Equivalence

For H, a Hopf coquasigroup, and A, a left quasi-H-module algebra, we show that the smash product A#H is linked to the algebra of H invariants AH by a Morita context. We use the Morita setting to prove that for finite dimensional H, there are equivalent conditions for A/AH to be Galois parallel in the case of H finite dimensional Hopf algebra.