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.

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

Definition

For categories $C, D$ (Category, object, morphism, domain, codomain, identity, composition, and hom-collection), a covariant functor $F : C \to D$ assigns an object $F A$ to every object $A$ and a morphism $F f : F A \to F B$ to every $f : A \to B$ , satisfying

$F (1_{A}) = 1_{F A}, F (g \circ f) = F g \circ F f .$

The identity functor acts identically on objects and morphisms. For $F : C \to D$ and $G : D \to E$ , the composite functor $G F$ has $(G F) A = G (F A)$ and $(G F) f = G (F f)$ .

A contravariant functor from $C$ to $D$ means a covariant functor $C^{o p} \to D$ , using Opposite category $C^{o p}$ .

Depends on

Dependency tree · 2 levels

Sources

Emily Riehl, Category Theory in Context, Chapter 1