TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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The unit is self-dual

Statement

In any monoidal category, the tensor unit $1$ is both a left dual and a right dual of itself.

Facts & Assumptions

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

[L1]

The two unitors agree on the unit object: $λ_{1} = ρ_{1}$ (The two unitors agree on the tensor unit).

[L2]

A left or right self-duality of $1$ requires an evaluation $1 \otimes 1 \to 1$ and a coevaluation $1 \to 1 \otimes 1$ satisfying the corresponding zig-zag identities (Left dual and right dual object).

Proof

technique · direct

1.1

Take the evaluation to be $λ_{1} : 1 \otimes 1 \to 1$ and the coevaluation to be $λ_{1}^{- 1} : 1 \to 1 \otimes 1$ . By [L1], this is the same pair as $ρ_{1}$ and $ρ_{1}^{- 1}$ .

givenL1L2construct

2.1

Substituting these maps into either zig-zag composite gives an instance of the triangle identity with every object equal to $1$ , so each composite is the identity of $1$ .

step 1.1algebra

3.1

Therefore $1$ is a left dual of itself, and because the same maps also satisfy the mirrored unit equations, it is a right dual of itself as well.

step 2.1L2∎

Depends on

Used by

Left duality is a contravariant antimonoidal functor Theorem

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, Section 2.10 (standard reference, not scraped)