  We construct a structure of a ring with local units on a co-Frobenius
coalgebra. We study a special class of co-Frobenius coalgebras whose objects we
call symmetric coalgebras. We prove that any semiperfect coalgebra can be
embedded in a symmetric coalgebra. A dual version of Brauer's equivalence
theorem is presented, allowing a characterization of symmetric coalgebras by
comparing certain functors. We define an automorphism of the ring with local
units constructed from a co-Frobenius coalgebra, which we call the Nakayama
automorphism. This is used to give a new characterization to symmetric
coalgebras and to describe Hopf algebras that are symmetric as coalgebras. As a
corollary we obtain as a consequence the known characterization of Hopf
algebras that are symmetric as algebras.
