CounterexampleConstruction: AI-generatedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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A componentwise family of morphisms need not be a natural transformation and hence need not be a unit

Statement refuted

For functors $F : C \to D$ and $G : D \to C$ , any family of correctly typed morphisms $η_{X} : X \to G F X$ is a possible unit.

Facts & Assumptions

Given: The identity functors $F = G = 1_{S e t}$ , the von Neumann sets $1 = {0}$ and $2 = {0, 1}$ , and the inclusion $i : 1 \to 2$ with $i (0) = 0$ .

[F1]

A natural transformation $α : F \Rightarrow G$ must satisfy $G f \circ α_{A} = α_{B} \circ F f$ for every $f : A \to B$ (Natural transformation and its components).

[F2]

Sets and functions form the large locally small category $S e t$ (Sets and functions form the large locally small category $S e t$ ).

[F3]

A unit of an adjunction is a natural transformation $1_{C} \Rightarrow G F$ (Adjunction by unit, counit, and the triangle identities).

Counterexample

technique · direct

1.1F2construct

Let $η_{2} : 2 \to 2$ transpose $0$ and $1$ , and let $η_{X} = 1_{X}$ for every set $X \neq 2$ . These are correctly typed components in the category [F2].

2.1step 1.1F1

For $i : 1 \to 2$ , the left side of [F1] is $i η_{1}$ , which sends $0$ to $0$ , while the right side is $η_{2} i$ , which sends $0$ to $1$ . Thus the naturality equation fails.

3.1step 2.1F3∎

The family is not a natural transformation and therefore cannot be a unit by [F3]. Correct component types alone do not suffice.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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)