  In this paper we call generalized lax epimorphism a functor defined on a ring
with several objects, with values in an abelian AB5 category, for which the
associated restriction functor is fully faithful. We characterize such a
functor with the help of a conditioned right cancellation of another,
constructed in a canonical way from the initial one. As consequences we deduce
a characterization of functors inducing an abelian localization and also a
necessary and sufficient condition for a morphism of rings with several objects
to induce an equivalence at the level of two localizations of the respective
module categories.
