PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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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

1.1

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.

1.2

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.

2.1

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

F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 16 results over 10 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., 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)