A note on Frobenius functors

Some new characterizations of Frobenius bimodules in terms of Frobenius functors were given. It was proved that a bimodule is Frobenius if and only if it is finitely generated projective on both sides, and that the restriction of the corresponding tensor functor to the categories of finitely generated projective modules is a Frobenius functor. The characterizations allow us to give a new proof of the endomorphism ring theorem by a functorial method.