CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge 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.

The two unitors agree on the tensor unit

Statement

In any monoidal category,

$λ_{1} = ρ_{1} .$

Facts & Assumptions

Given: A monoidal category with tensor unit $1$ .

[L1]

The left unitor satisfies $λ_{1 \otimes 1} = (λ_{1} \otimes 1_{1}) \circ α_{1, 1, 1}$ (The left unitor of a tensor product is determined by the associator).

[L2]

The right unitor satisfies $ρ_{1 \otimes 1} \circ α_{1, 1, 1} = 1_{1} \otimes ρ_{1}$ (The right unitor of a tensor product is determined by the associator).

[F1]

Corollary 2.2.5 of EGNO proves that, under the same monoidal-category axioms and associator convention, one has $λ_{1} = ρ_{1}$ .

Proof

technique · direct

1.1

The monoidal-category hypotheses required by [F1] are exactly the ones under which [L1] and [L2] were proved, so [F1] applies to the present category.

givenL1L2F1

2.1

Therefore $λ_{1} = ρ_{1}$ .

step 1.1F1

3.1

Therefore the two unitors agree on the tensor unit.

step 2.1∎

Depends on

Used by

Dependency tree · two levels

3 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, Corollary 2.2.5 (standard reference, not scraped)