In this chapter, the R-bialgebroids and Hopf algebroids of the previous chapter are investigated further in the particular case when the base algebra R carries a so-called separable Frobenius structure. Separable Frobenius structures on some algebra R are shown to correspond to separable Frobenius structures on the forgetful functor from the category of R-bimodules to the category of vector spaces. Based on that, the bijection of Chap. 5 is refined to bijections between three structures, for any algebra A. First, monoidal structures on the category of A-modules together with separable Frobenius structures on the forgetful functor to the category of vector spaces. Second, bialgebroid structures on A over some base algebra R, together with separable Frobenius structures on R. Finally, weak bialgebra structures on A. The R-bialgebroid A is a Hopf algebroid if and only if the corresponding weak bialgebra A is a weak Hopf algebra.