DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-16

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.

Mutually left and mutually right adjoint contravariant functors

Definition

Let $S : C^{o p} \to D$ and $T : D^{o p} \to C$ be contravariant functors (Covariant functor, identity functor, composite functor, and contravariant functor) between locally small categories (Small, locally small, and large categories). They are mutually left and mutually right adjoint when there are bijections

$D (d, S c) ≅ C (c, T d)$

natural in $c \in C$ and $d \in D$ . Equivalently, the covariant functors $S : C^{o p} \to D$ and $T^{o p} : D \to C^{o p}$ form an adjoint pair, and the same data give the opposite adjunction after reversing both categories.

Depends on

Used by

Vanishing sets and vanishing ideals form a contravariant Galois connection Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 9 results over 5 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, 2nd ed., Definition 4.4.1 (standard reference, not scraped)