DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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.

Identity natural transformation and vertical composition

Definition

For a functor $F : C \to D$ , the identity natural transformation $1_{F} : F \Rightarrow F$ has component $(1_{F})_{A} = 1_{F A}$ .

For natural transformations $α : F \Rightarrow G$ and $β : G \Rightarrow H$ (Natural transformation and its components), their vertical composite $β \circ α : F \Rightarrow H$ is defined componentwise by

$(β \circ α)_{A} = β_{A} \circ α_{A} .$

The fact that this componentwise family is natural is discharged by Vertical composites of natural transformations satisfy naturality ↗.

Depends on

Natural transformation and its components

Used by

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)