DefinitionDefinition: AI-adaptedProof: 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 isomorphism

Definition

A natural isomorphism $α : F \Rightarrow G$ is a natural transformation for which there is a natural transformation $β : G \Rightarrow F$ with $β \circ α = 1_{F}$ and $α \circ β = 1_{G}$ . The compositions here are the vertical compositions of Identity natural transformation and vertical composition.

When the source category is small, so that Functor category $[C, D]$ is formed, this says exactly that $α$ is an isomorphism from $F$ to $G$ in that functor category. Thus the definition combines Natural transformation and its components with the categorical notion of isomorphism from Isomorphism, groupoid, and connected category without requiring a functor category for an arbitrary large source.

Depends on

Dependency tree · 2 levels

Sources

Emily Riehl, Category Theory in Context, Chapter 1