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.

Whiskering and horizontal composition of natural transformations

Definition

Let $α : F \Rightarrow G : C \to D$ be a natural transformation (Natural transformation and its components), and let $H : D \to E$ and $K : B \to C$ be functors (Covariant functor, identity functor, composite functor, and contravariant functor). The left whiskering $H α : H F \Rightarrow H G$ has components $H (α_{A})$ , and the right whiskering $α K : F K \Rightarrow G K$ has components $α_{K B}$ .

For $β : H \Rightarrow L : D \to E$ , the horizontal composite $β * α : H F \Rightarrow L G$ has either equal component formula

$(β * α)_{A} = β_{G A} \circ H (α_{A}) = L (α_{A}) \circ β_{F A} .$

Their equality is the naturality equation for $β$ at $α_{A}$ . Naturality of the resulting family is proved in Horizontal composites of natural transformations satisfy naturality ↗.

Depends on

Used by

Dependency tree · two levels

4 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)