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 adjunction restricts to an equivalence on the subcategories fixed by its unit and counit

Statement

For an adjunction $F ⊣ G$ , let $C_{η}$ be the full subcategory of objects $c$ for which $η_{c}$ is an isomorphism, and let $D_{ε}$ be the full subcategory of objects $d$ for which $ε_{d}$ is an isomorphism. Then $F$ and $G$ restrict to an adjoint equivalence

$C_{η} ≃ D_{ε} .$

Facts & Assumptions

Given: An adjunction $F ⊣ G$ with unit $η$ and counit $ε$ .

[F1]

A full subcategory contains chosen objects and all morphisms between them from the ambient category (Subcategory and full subcategory).

[F2]

An equivalence consists of quasi-inverse functors and natural isomorphisms between their composites and the identity functors (Equivalence, quasi-inverse, and adjoint equivalence of categories).

[L1]

The triangle identities are $ε_{F c} F (η_{c}) = 1_{F c}$ and $G (ε_{d}) η_{G d} = 1_{G d}$ (Adjunction by unit, counit, and the triangle identities).

Proof

technique · direct

1.1

If $η_{c}$ is invertible, then $F (η_{c})$ is invertible, and the first triangle identity makes $ε_{F c} = F (η_{c})^{- 1}$ ; hence $F c \in D_{ε}$ .

1.2

Dually, if $ε_{d}$ is invertible, then the second triangle identity makes $η_{G d} = G (ε_{d})^{- 1}$ ; hence $G d \in C_{η}$ .

2.1

By [F1], $F$ and $G$ therefore restrict to functors between the two full subcategories, and the restricted unit and counit remain natural.

step 1.1step 1.2F1

3.1

Every component of the restricted unit and counit is an isomorphism by definition, so [F2] makes the restrictions quasi-inverse equivalences; the inherited triangle identities make the equivalence adjoint.

step 2.1F2L1∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 14 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., Lemma 4.2.11 (standard reference, not scraped)