  Let $\Cc$ and $\Dd$ be two corings over a ring $A$ and
$\Cc\stackrel{\lambda}{\longrightarrow}\Dd$ be a morphism of corings. We
investigate the situation when the associated induced ("corestriction of
scalars") functor $\Mm^\Cc\longrightarrow \Mm^\Dd$ is a Frobenius functor, and
call these morphisms Frobenius extensions of corings. The characterization
theorem generalizes notions such as Frobenius corings and is applied to several
situations; in particular, provided some (general enough) flatness conditions
hold, the notion proves to be dual to that of Frobenius extensions of rings
(algebras). Several finiteness theorems are given for each case we consider;
these theorems extend existing results from Frobenius extensions of rings or
from Frobenius corings, showing that a certain finiteness property almost
always occur for many instances of Frobenius functors.
