TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

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Every equivalence of categories can be equipped as an adjoint equivalence

Statement

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

Facts & Assumptions

Given: Equivalence data $F : C ⇄ D : G$ with a natural isomorphism $η : 1_{C} \Rightarrow G F$ .

[L1]

An adjoint equivalence is equivalence data satisfying two triangle identities (Equivalence, quasi-inverse, and adjoint equivalence of categories), and natural isomorphisms have invertible components (A natural transformation is a natural isomorphism exactly when every component is an isomorphism).

[L2]

Every equivalence is fully faithful by A functor is an equivalence exactly when it is fully faithful and split essentially surjective, without Choice, and fully faithful functors reflect isomorphisms (Every fully faithful functor reflects isomorphisms).

Proof

technique · direct

1.1

The quasi-inverse $G$ is itself an equivalence, hence fully faithful by [L2]; for each $D$ , fullness gives a unique $ε_{D}^{'} : F G D \to D$ with $G (ε_{D}^{'}) = η_{G D}^{- 1}$ .

givenL1L2

2.1

Naturality of $η$ and faithfulness of $G$ show that the components $ε_{D}^{'}$ are natural; since $G (ε_{D}^{'})$ is an isomorphism, reflection in [L2] makes each $ε_{D}^{'}$ an isomorphism.

step 1.1L1L2

3.1

The defining equation gives $G ε^{'} \circ η G = 1_{G}$ ; applying $G$ to $ε^{'} F \circ F η$ and using naturality of $η$ at each $η_{A}$ gives the identity, and faithfulness of $G$ yields $ε^{'} F \circ F η = 1_{F}$ .

step 2.1L1L2

4.1

Thus $(F, G, η, ε^{'})$ satisfies both triangle identities and is an adjoint equivalence.

step 3.1L1∎

Depends on

Used by

Monoidal equivalence and monoidal quasi-inverse data Definition
An adjoint equivalence is an adjunction whose unit and counit are natural isomorphisms Proposition
Mac Lane strictification Theorem

Dependency tree · two levels

10 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, Chapter 1 (standard reference, not scraped)
Ahrens, Kapulkin and Shulman, Univalent categories and the Rezk completion, section 6 (standard reference, not scraped)