  Firm Frobenius algebras are firm algebras and counital coalgebras such that
the comultiplication is a bimodule map. They are investigated by categorical
methods based on a study of adjunctions and lifted functors. Their categories
of comodules and of firm modules are shown to be isomorphic if and only if a
canonical comparison functor from the category of comodules to the category of
non-unital modules factorizes through the category of firm modules. This
happens for example if the underlying algebra possesses local units, e.g. the
firm Frobenius algebra arises from a co-Frobenius coalgebra over a base field;
or if the comultiplication splits the multiplication (hence the underlying
coalgebra is coseparable).
