LemmaStatement: 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.

Vertical composites of natural transformations satisfy naturality

Statement

The vertical composite of two natural transformations satisfies the naturality equation.

Facts & Assumptions

Given: Natural transformations $α : F \Rightarrow G$ and $β : G \Rightarrow H$ , and a morphism $f : A \to B$ .

[L1]

Vertical composition is componentwise, and $α, β$ satisfy their naturality equations (Identity natural transformation and vertical composition).

Proof

technique · direct

1.1

By naturality, $G f \circ α_{A} = α_{B} \circ F f$ and $H f \circ β_{A} = β_{B} \circ G f$ .

givenL1

2.1

Therefore $H f \circ (β_{A} \circ α_{A}) = (β_{B} \circ G f) \circ α_{A} = β_{B} \circ (α_{B} \circ F f) = (β_{B} \circ α_{B}) \circ F f$ .

step 1.1L1

3.1

This is the naturality equation for the componentwise family $β \circ α$ .

step 2.1L1∎

Depends on

Identity natural transformation and vertical composition

Used by

Cited to discharge well-definedness by Identity natural transformation and vertical composition.

Dependency tree · two levels

2 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)