CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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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.

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A componentwise family between functors need not be a natural transformation

Statement refuted

Choosing one morphism between each pair of object values of two functors does not automatically give a natural transformation.

Facts & Assumptions

Given: The walking-arrow category $C=(0\xrightarrow{a}1)$ and the category $\mathbf{Set}$ .

[L1]

Naturality requires $G(a)\eta_0=\eta_1F(a)$ for the arrow $a$ (Natural transformation and its components).

[L2]

Sets and functions form $\mathbf{Set}$ (Sets and functions form the large locally small category $\mathbf{Set}$ ).

Counterexample

technique · direct

1.1

Let $F:C\to\mathbf{Set}$ be constant at the singleton $\{*\}$ and let $G:C\to\mathbf{Set}$ be constant at $\{0,1\}$ , with both functors sending $a$ to the relevant identity function.

2.1

Define components by $\eta_0(*)=0$ and $\eta_1(*)=1$ . Each is a valid function $F(i)\to G(i)$ .

step 1.1L2

3.1

At $a$ , the left side of the naturality equation sends $*$ to $0$ , whereas the right side sends $*$ to $1$ . Thus $G(a)\eta_0\ne\eta_1F(a)$ .

step 1.1step 2.1L1

4.1

The family $(\eta_0,\eta_1)$ has a component of the correct type at every object but fails naturality.

step 3.1L1∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 19 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.