  We investigate left and right co-Frobenius coalgebras and give equivalent
characterizations which prove statements dual to the characterizations of
Frobenius algebras. We prove that a coalgebra is left and right co-Frobenius if
and only if C is isomorphic to Rat(C*) as right C*-modules and also that this
is eqhivalent to the fact that the functors Hom_K(-,K) and Hom_{C*}(-,C*) from
the category of right C-comodules to the category of all left C*-modules are
isomorphic. This allows a definition of a left-right symmetric concept of
co-Frobenius coalgebras that is perfectly dual to the one of Frobenius algebras
and coincides to the existing notion left and right co-Frobenius coalgebra.
