Coproductid

Coproductid is an informal term used to denote the data that identifies a particular coproduct object in a category. Given a small family of objects {A_i} in a category C, a coproduct consists of an object ⊔_i A_i together with injection morphisms in_i: A_i → ⊔_i A_i that satisfy the universal property: for any family of morphisms f_i: A_i → B, there exists a unique morphism f: ⊔_i A_i → B with f ∘ in_i = f_i. The coproductid refers to this chosen coproduct object along with its injections, i.e., the triple (⊔_i A_i, {in_i}).

Because coproducts are unique up to isomorphism, multiple isomorphic choices exist. The coproductid serves as a concrete representative needed for constructions, computations, or functorial operations that require a fixed object and maps.

Examples: In Set, the coproduct of A_i is the disjoint union with the canonical injections; in Ab, the coproduct of a family of abelian groups is the direct sum with the natural inclusions.

Notes: coproductid is not a standard technical term; when encountered, it should be interpreted as the selected coproduct data (object and injections) for a particular index family. In general, the concept aligns with the broader idea of specifying a colimit with a fixed choice of representative.