An additive category is an abstraction of the properties of the category of modules. In such a category finite products and coproducts agree and the direct sum is either of them, cf. biproduct. General case: In category theory the direct sum is often, but not always, the coproduct in the category of the mathematical objects in question. For example, in the category of abelian groups, direct sum is a coproduct. This is also true in the category of modules. The direct sum formula_24 comes equipped with a "projection" homomorphism formula_67 for each "j" in "I" and a "coprojection" formula_68 for each "j" in "I". Given another algebraic structure formula_5 (with the same additional structure) and homomorphisms formula_70 for every "j" in "I", there is a unique homomorphism formula_71, called the sum of the "g""j", such that formula_72 for all "j". Thus the direct sum is the coproduct in the appropriate category.