TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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 $F:\mathcal C\rightleftarrows\mathcal D:G$ with a natural isomorphism $\eta:1_{\mathcal C}\Rightarrow GF$ .

[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 $\varepsilon'_D:FGD\to D$ with $G(\varepsilon'_D)=\eta_{GD}^{-1}$ .

givenL1L2

2.1

Naturality of $\eta$ and faithfulness of $G$ show that the components $\varepsilon'_D$ are natural; since $G(\varepsilon'_D)$ is an isomorphism, reflection in [L2] makes each $\varepsilon'_D$ an isomorphism.

step 1.1L1L2

3.1

The defining equation gives $G\varepsilon'\circ\eta G=1_G$ ; applying $G$ to $\varepsilon'F\circ F\eta$ and using naturality of $\eta$ at each $\eta_A$ gives the identity, and faithfulness of $G$ yields $\varepsilon'F\circ F\eta=1_F$ .

step 2.1L1L2

4.1

Thus $(F,G,\eta,\varepsilon')$ satisfies both triangle identities and is an adjoint equivalence.

step 3.1L1∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 9 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, Chapter 1 (standard reference, not scraped)
Ahrens, Kapulkin and Shulman, Univalent categories and the Rezk completion, section 6 (standard reference, not scraped)