DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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.

Monoidal natural transformation

Definition

Let $F, G : C \to D$ be lax monoidal functors (Lax, strong, and strict monoidal functors). A monoidal natural transformation $θ : F \Rightarrow G$ is a natural transformation (Natural transformation and its components) such that

$θ_{X \otimes Y} \circ F_{2; X, Y} = G_{2; X, Y} \circ (θ_{X} ⊠ θ_{Y})$

for all objects $X, Y$ , and

$G_{0} = θ_{1} \circ F_{0} .$

Thus $θ$ is compatible both with the binary structure maps and with the unit map.

Depends on

Used by

Dependency tree · two levels

5 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

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Definition 2.4.8 (standard reference, not scraped)