Zagier duality and integrality of Fourier coefficients for weakly holomorphic modular forms

In this note, we first see that the isomorphism in [26] between vector-valued modular forms and scalar-valued modular forms holds for more general discriminant forms. With this established, we shall prove the Zagier duality for canonical bases for spaces of weakly holomorphic modular forms satisfying some ϵ-condition, thus obtain the complete grids for the duality. Finally we raise a question on certain integrality of the Fourier coefficients of reduced modular forms and provide a partial solution.