Good point, I did not see that! So the whole thing boils down to a very simple observation on pointed endofunctors (which is certainly well-known for reflectors): suppose you have an endofunctor F and a natural transformation u:Id --> F which is pointwise monic; then, if Fu is pointwise epic, u itself is pointwise epic. Proof: u_A x,y A ---> FA ---> B | | | u_A | u_FA | | u_B | | | FA --> FFA --> FB Fu_A Fx,Fy ( xu = yu gives Fx.Fu = Fy.Fu, hence Fx = Fy and then ux = uy and x = y.) Coming back to injectivity: if u was pointwise an H-injective hull, then both uF and Fu must be isos (independently of H being a class of monos or not!). Hence, if H is a class of monos, the statement above applies.
Walter wrote "We are able to compensate for the loss of mono through condition 1". Wouldn't it be simpler just to say that condition 1 implies that everything in H is a monomorphisms? x y (Let A --> B be an H-morphism and let X --> A, X --> A be such x y that X --> A --> B = X --> A --> B. If x were different from y then there would be A --> E, E an H-injective object, so that x y X --> A --> E were different from X --> A --> E. But there would have to be B --> E such that A --> E = A --> B --> E and x y X --> A --> B would have be different from X --> A --> B.)
So H-morphism is a strengthening of monic and that put's us back to the situation I outlined:
If the strengthening of monic is such that it becomes an iso whenever epic then there's an easy proof of the impossibility of functoriality, with or without a cogenerator.
-- End of excerpt from Peter Freyd