DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-11

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.

Natural transformation and its components

Let $F, G : C \to D$ be functors (Covariant functor, identity functor, composite functor, and contravariant functor). A natural transformation $α : F \Rightarrow G$ is a family of morphisms $α_{A} : F A \to G A$ , one for each object $A$ of $C$ , such that every $f : A \to B$ satisfies the naturality equation

$G f \circ α_{A} = α_{B} \circ F f .$

The morphism $α_{A}$ is the component of $α$ at $A$ .

Depends on