False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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A natural transformation is determined by its component at one object

Statement

FALSE. A natural transformation between two functors is determined by its component at any one object of the source category.

Facts & Assumptions

Given: The discrete category $C$ on two objects $0,1$ and the category $\mathbf{Set}$ .

[L1]

A natural transformation is a component family constrained by a naturality equation for each source morphism (Natural transformation and its components).

[L2]

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

Refutation

technique · direct

1.1

Let $F=G:C\to\mathbf{Set}$ send both objects to $S=\{0,1\}$ and each identity to $1_S$ . Because $C$ is discrete, any two functions $S\to S$ chosen as components satisfy all naturality equations.

L1L2

2.1

Let $\alpha_0=\alpha_1=1_S$ . Let $\beta_0=1_S$ and let $\beta_1$ transpose $0$ and $1$ . Step 1.1 makes both $\alpha$ and $\beta$ natural transformations $F\Rightarrow G$ .

step 1.1

3.1

The transformations agree at object $0$ because $\alpha_0=\beta_0$ , but they differ at object $1$ because $\alpha_1\ne\beta_1$ .

step 2.1

4.1

Therefore one component does not determine a natural transformation when the source category has an unrelated component.

step 3.1∎

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: 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.

Sources

Saunders Mac Lane, Categories for the Working Mathematician, Chapter II (standard reference, not scraped)