Some comments on:
Theorem. Let H be a class of morphisms in a category C such that 1. all H-injective objects form a cogenerating class, and 2. the class of all H-essential morphisms which are epimorphic is precisely the class of isomorphisms of C . Then C cannot have natural H-injective hulls (i.e. they cannot form an endofunctor together with a natural transformation from Id) unless every object in C is H-injective.
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.