False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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A unit and counit determine an adjunction without the triangle identities

Statement

If functors $F : C \to D$ and $G : D \to C$ admit natural transformations $η : 1_{C} \Rightarrow G F$ and $ε : F G \Rightarrow 1_{D}$ , then $F ⊣ G$ even when the triangle identities have not been checked.

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 in that category 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 natural transformations $η$ and $ε$ satisfying $(ε F) (F η) = 1_{F}$ and $(G ε) (η G) = 1_{G}$ (Adjunction by unit, counit, and the triangle identities).

Refutation

technique · direct

1.1F1construct

Regard $C_{2}$ as the one-object category $C$ supplied by [F1], and take $F = G = 1_{C}$ .

2.1step 1.1algebra

Let the sole component of $η$ be $1$ and the sole component of $ε$ be $z$ . Both are natural because $C_{2}$ is abelian, so each component commutes with every morphism.

3.1step 2.1F2algebra∎

Each triangle composite is $z 1 = z$ , which is not the identity morphism $1$ . Thus the data fail both identities in [F2] and do not form an adjunction.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

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