PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)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.

An adjoint equivalence is an adjunction whose unit and counit are natural isomorphisms

Statement

For functors $F : C \to D$ and $G : D \to C$ , the following data are equivalent:

an adjoint equivalence between $C$ and $D$ with functors $F$ and $G$ ;
an adjunction $F ⊣ G$ (Adjunction by unit, counit, and the triangle identities) whose unit and counit are natural isomorphisms.

Consequently, every equivalence of categories can be equipped with such an adjunction.

Facts & Assumptions

Given: Categories $C, D$ and functors $F : C \to D$ , $G : D \to C$ .

[F1]

An adjoint equivalence consists of $F, G$ , a natural isomorphism $η : 1_{C} \Rightarrow G F$ , a natural isomorphism $ε : F G \Rightarrow 1_{D}$ , and the two triangle identities (Equivalence, quasi-inverse, and adjoint equivalence of categories).

[F2]

Every equivalence of categories admits a choice of unit and counit satisfying the triangle identities, and hence can be equipped as an adjoint equivalence (Every equivalence of categories can be equipped as an adjoint equivalence).

Proof

technique · direct

1.1F1

Starting with an adjoint equivalence, forget only the assertion that $η$ and $ε$ are invertible. The remaining functors, natural transformations, and triangle identities are an adjunction, while the forgotten assertion still says its unit and counit are natural isomorphisms.

1.2F1

Conversely, an adjunction with invertible unit and counit has exactly the functors, natural isomorphisms, and triangle identities required by [F1], so it is an adjoint equivalence.

2.1F2step 1.1∎

Finally, [F2] equips any equivalence with the data in step 1.1, proving the consequence.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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, 2nd ed., Proposition 4.3.5 (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Section 1.3 and Theorem 2.2.5 (standard reference, not scraped)