On the isomorphism of injective objects in Grothendieck categories

Recently, some of the authors of the present note provided a generalization of Bumby's theorem to the injectivity of modules over some classes of monomorphisms. The question of whether Bumby's criterion could be extended to more general classes of categories was left without an affirmative response, though. However, the present note provides an extension of that theorem to Grothendieck categories. More concretely, we establish that any two injective objects in a Grothendick category are isomorphic when they are isomorphic to subobjects of each other. As a corollary, we prove that two objects have isomorphic injective envelops (if they exist) whenever each of them is isomorphic to a subobject of the other. In the way, it will be indispensable to note that the main results of our previous work are still valid under weaker conditions.Keywords: Grothendiek categories, Bumby's theorem, isomorphism criterion, injective objects, injective envelop