Faithfulness and the coequalizer of the kernel pair process

Starting from a ground structure consisting of an adjunction C ! X and a pref- actorization system (E;M) which factorizes the unit morphisms 'C : C ! GF(C), C 2 C, through an epimorphism ·C, a full epire∞ection C ! M is obtained with unit · : 1C ! HI. A re∞ective factorization system is associated, and there may be a concordant-dissonant (in a similar sense to the one referred in (3)) and also a monotone-light factorization. We will show that in the case of any adjunction SetA ! SetB, given by right Kan extensions along a functor K : B ! A, there is a monotone-light factorization which coincides with the concordant-dissonant one, pro- vided the objects in the image of the functor K are a cogenerating set for A. Remark that this condition is equivalent to demanding that the composition SetK ¢y with the Yoneda embedding is a faithful functor. A generalization of the given results, to left adjoints from presheaves into a cocomplete category, is then possible.