CounterexampleConstruction: AI-generatedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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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.
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A wrong counit can be natural while both triangle identities fail

Statement refuted

For identity functors, any natural choices of unit and counit give the identity adjunction.

Facts & Assumptions

Given: The two-element group $C_{2} = {1, z}$ with $z^{2} = 1$ .

[F1]

Every monoid is a one-object category, and it is a group exactly when every morphism is invertible (A monoid is a one-object category, and a group is a one-object category in which every morphism is invertible).

[F2]

An adjunction requires $(ε F) (F η) = 1_{F}$ and $(G ε) (η G) = 1_{G}$ (Adjunction by unit, counit, and the triangle identities).

Counterexample

technique · direct

1.1

Regard $C_{2}$ as a one-object category $C$ by [F1] and set $F = G = 1_{C}$ . Choose the identity element as the component of $η$ and $z$ as the component of $ε$ .

F1construct

2.1

Both transformations are natural because $C_{2}$ is abelian, but each triangle composite is $z 1 = z \neq 1$ . Hence both identities in [F2] fail.

step 1.1F2algebra

3.1

Replacing $ε$ by the identity element makes both composites equal to $1$ , recovering the identity adjunction and isolating the failure in the wrong counit.

step 2.1F2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 17 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Emily Riehl, Category Theory in Context, 2nd ed., Definition 4.2.1 (standard reference, not scraped)