DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-08-27

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Canonical morphism from a finite coproduct to a finite product

Definition

Let $C$ have zero morphisms, and let $(A_{i})_{i \in I}$ be a finite family of objects with coproduct $(Q, ι_{i})$ and product $(P, p_{i})$ (Products and coproducts as limits and colimits of discrete diagrams, including their existence-and-uniqueness equations).

The canonical morphism from the finite coproduct to the finite product is the unique morphism

$c : Q \to P$

whose components satisfy

$p_{j} c ι_{i} = {\begin{cases} 1_{A_{i}}, & i = j, \\ 0_{A_{i}, A_{j}}, & i \neq j . \end{cases}$

The zero morphisms used here are the canonical ones supplied by A preadditive category with a zero object has zero morphisms in the published sense when the ambient category is preadditive with a zero object.

Depends on

Used by

Biproduct Definition
FALSE: finite products and finite coproducts already force biproducts False statement

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Section 12.3: Preadditive and additive categories (standard reference, not scraped)