False statementConstruction: AI-adaptedVerification: AI-generatedprecheck 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 $S e t$ .

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

Refutation

technique · direct

1.1

Let $F = G : C \to S e t$ 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 $α_{0} = α_{1} = 1_{S}$ . Let $β_{0} = 1_{S}$ and let $β_{1}$ transpose $0$ and $1$ . Step 1.1 makes both $α$ and $β$ natural transformations $F \Rightarrow G$ .

step 1.1

3.1

The transformations agree at object $0$ because $α_{0} = β_{0}$ , but they differ at object $1$ because $α_{1} \neq β_{1}$ .

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 · 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.

Sources

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