PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A natural transformation is a natural isomorphism exactly when every component is an isomorphism

Statement

A natural transformation $α : F \Rightarrow G$ is a natural isomorphism exactly when every component $α_{A}$ is an isomorphism.

Facts & Assumptions

Given: A natural transformation $α : F \Rightarrow G$ .

[L1]

A natural isomorphism has a two-sided inverse natural transformation (Natural isomorphism), and vertical composition and identity transformations are componentwise (Identity natural transformation and vertical composition).

Proof

technique · direct

1.1

If $α$ has a natural inverse $β$ , then $β_{A} α_{A} = 1_{F A}$ and $α_{A} β_{A} = 1_{G A}$ at every object, so each $α_{A}$ is an isomorphism.

givenL1

2.1

Conversely, suppose every $α_{A}$ is invertible and put $β_{A} = α_{A}^{- 1}$ ; from $G f α_{A} = α_{B} F f$ , composition with the two inverses gives $F f β_{A} = β_{B} G f$ , so $β : G \Rightarrow F$ is natural.

step 1.1L1

3.1

Componentwise, $β \circ α = 1_{F}$ and $α \circ β = 1_{G}$ , hence $α$ is a natural isomorphism.

step 2.1L1∎

Depends on

Used by

Dependency tree · two levels

5 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, Chapter 1 (standard reference, not scraped)