Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedSession-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.

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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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A natural transformation is a natural isomorphism exactly when every component is an isomorphism

Statement

A natural transformation α:F⇒G\alpha:F\Rightarrow G is a natural isomorphism exactly when every component αA\alpha_A is an isomorphism.

Facts & Assumptions

Given: A natural transformation α:F⇒G\alpha: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\alpha has a natural inverse β\beta, then βAαA=1FA\beta_A\alpha_A=1_{FA} and αAβA=1GA\alpha_A\beta_A=1_{GA} at every object, so each αA\alpha_A is an isomorphism.

givenL1
2.1

Conversely, suppose every αA\alpha_A is invertible and put βA=αA−1\beta_A=\alpha_A^{-1}; from GfαA=αBFfGf\alpha_A=\alpha_BFf, composition with the two inverses gives FfβA=βBGfFf\beta_A=\beta_BGf, so β:G⇒F\beta:G\Rightarrow F is natural.

step 1.1L1
3.1

Componentwise, β∘α=1F\beta\circ\alpha=1_F and α∘β=1G\alpha\circ\beta=1_G, hence α\alpha is a natural isomorphism.

step 2.1L1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 9 results over 8 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