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

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.

A left dual of an object has that object as a right dual

Statement

If $X^{\lor}$ is a left dual of an object $X$ , then $X$ is a right dual of $X^{\lor}$ . Dually, if $^{\lor} X$ is a right dual of $X$ , then $X$ is a left dual of $^{\lor} X$ .

Facts & Assumptions

Given: A left dual $(X^{\lor}, {ev}_{X}, {coev}_{X})$ of $X$ .

[L1]

A right dual of $X^{\lor}$ requires maps $X^{\lor} \otimes X \to 1$ and $1 \to X \otimes X^{\lor}$ satisfying the mirror zig-zag identities (Left dual and right dual object, The zig-zag identities).

Proof

technique · direct

1.1

Reuse the same two morphisms ${ev}_{X} : X^{\lor} \otimes X \to 1$ and ${coev}_{X} : 1 \to X \otimes X^{\lor}$ , but now regard them as candidate right-dual data for the object $X^{\lor}$ with proposed right dual $X$ .

givenL1construct

2.1

The first right-dual zig-zag for $X^{\lor}$ is exactly the second left-dual zig-zag for $X$ , and the second right-dual zig-zag for $X^{\lor}$ is exactly the first left-dual zig-zag for $X$ . Both are identities by the hypotheses that $X^{\lor}$ is a left dual of $X$ .

step 1.1L1

3.1

Hence $X$ is a right dual of $X^{\lor}$ . The dual assertion is the same argument with left and right interchanged.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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, Remark 2.10.3 (standard reference, not scraped)
Michael Muger, Tensor Categories: A Selective Guided Tour, Section 1.5 (standard reference, not scraped)