On the equivalence of Batyrev and BHK mirror symmetry constructions

We consider the connection between two constructions of the mirror partner for the Calabi-Yau orbifold. This orbifold is defined as a quotient by some suitable subgroup G of the phase symmetries of the hypersurface X_M in the weighted projective space, cut out by a quasi-homogeneous polynomial W_M. The first, Berglund-Hübsch-Krawitz (BHK) construction, uses another weighted projective space and the quotient of a new hypersurface X_M^T inside it by some dual group G^T. In the second, Batyrev construction, the mirror partner is constructed as a hypersurface in the toric variety defined by the reflexive polytope dual to the polytope associated with the original Calabi-Yau orbifold. We give a simple evidence of the equivalence of these two constructions.