Co-Frobenius coalgebras

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 as right C∗-modules and also that this is equivalent to the fact that the functors HomK(−,K) and HomC∗(−,C∗) from MC to 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.