  A pair of adjoint functors $(F,G)$ is called a Frobenius pair of the second
type if $G$ is a left adjoint of $\beta F\alpha$ for some category equivalences
$\alpha$ and $\beta$. Frobenius ring extensions of the second kind provide
examples of Frobenius pairs of the second kind. We study Frobenius pairs of the
second kind between categories of modules, comodules, and comodules over a
coring. We also show that a finitely generated projective Hopf algebra over a
commutative ring is always a Frobenius extension of the second kind, and prove
that the integral spaces of the Hopf algebra and its dual are isomorphic.
