Filtered products of injective objects in Grothendieck categories

The transfer of properties from a family of modules to their direct product or to their direct sum has always been a topical subject of great interest. In this paper we address this type of questions in the general framework of (locally finitely generated) Grothendieck categories and for more general constructions that leave as particular cases direct sums or products: F -products, F being a filter on the chosen index set. Thus, we study the conditions on the categories for the F -products of any family of (M-)injective objects, or else of any family of copies of a single injective object, to be injective.