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

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Reversing the tensor product exchanges left and right duals

Statement

Let $C^{r e v}$ be the reverse monoidal category of $C$ . An object is a left dual of $X$ in $C^{r e v}$ if and only if it is a right dual of $X$ in $C$ , and similarly with "left" and "right" interchanged.

Facts & Assumptions

Given: A monoidal category $C$ and an object $X$ of $C$ .

[L1]

In the reverse monoidal category, the tensor order is reversed and the unitors are swapped: $λ_{X}^{r e v} = ρ_{X}$ and $ρ_{X}^{r e v} = λ_{X}$ (The reverse and the opposite of a monoidal category).

[L2]

Left and right duality are defined by the explicit zig-zag composites in Left dual and right dual object.

Proof

technique · direct

1.1

Suppose $Y$ is a left dual of $X$ in $C^{r e v}$ , with evaluation $Y \otimes^{r e v} X = X \otimes Y \to 1$ and coevaluation $1 \to X \otimes^{r e v} Y = Y \otimes X$ .

givenL1L2

2.1

Writing the two left-dual zig-zag composites in $C^{r e v}$ and then translating them with [L1] replaces $\otimes^{r e v}$ by reversed tensor order, $α^{r e v}$ by $α^{- 1}$ , and the reverse unitors by the ordinary opposite ones. The result is exactly the pair of right-dual zig-zag composites for $Y$ as a right dual of $X$ in $C$ .

step 1.1L1L2

3.1

Therefore the left-dual axioms in $C^{r e v}$ are equivalent to the right-dual axioms in $C$ . The converse and the left/right-swapped statement are the same calculation in reverse.

step 2.1∎

Depends on

Used by

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Sources

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Section 2.10 (standard reference, not scraped)