CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck 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 $C = (0 \overset{a}{\to} 1)$ and the category $S e t$ .

[L1]

Naturality requires $G (a) η_{0} = η_{1} F (a)$ for the arrow $a$ (Natural transformation and its components).

[L2]

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

Counterexample

technique · direct

1.1

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

2.1

Define components by $η_{0} (*) = 0$ and $η_{1} (*) = 1$ . 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 $G (a) η_{0} \neq η_{1} F (a)$ .

step 1.1step 2.1L1

4.1

The family $(η_{0}, η_{1})$ has a component of the correct type at every object but fails naturality.

step 3.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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.