DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 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.

Natural transformation and its components

Definition

Let $F,G:\mathcal C\to\mathcal D$ be functors (Covariant functor, identity functor, composite functor, and contravariant functor). A natural transformation $\alpha:F\Rightarrow G$ is a family of morphisms $\alpha_A:FA\to GA$ , one for each object $A$ of $\mathcal C$ , such that every $f:A\to B$ satisfies the naturality equation

$Gf\circ\alpha_A=\alpha_B\circ Ff.$

The morphism $\alpha_A$ is the component of $\alpha$ at $A$ .

Depends on

Covariant functor, identity functor, composite functor, and contravariant functor

Used by

A componentwise family between functors need not be a natural transformation Counterexample
Functor category [mathcal C,mathcal D] Definition
Identity natural transformation and vertical composition Definition
Natural isomorphism Definition
Whiskering and horizontal composition of natural transformations Definition
For n≥ 1, determinant is a natural transformation det:GLₙ(-)⟹(-)^× from commutative rings to groups Example
Singletons define a natural transformation from the identity functor on sets to the covariant power-set functor Example
A natural transformation is determined by its component at one object False statement

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 5 results over 4 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

Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)