LAUSR

For H a Hopf quasigroup and C, a left quasi H-comodule coalgebra, we show that the smash coproduct C⋊H (as a symmetry of smash product) is linked to some quotient coalgebra Q=C/CH*+ by a Morita-Takeuchi context (as a symmetry of Morita context). We use the Morita-Takeuchi setting to prove that for finite dimensional H, equivalent conditions for C/Q to be a Hopf quasigroup Galois coextension (as a symmetry of Galois extension). In particular, we consider a special case of quasigroup graded coalgebras as an application of our theory.