TheoremStatement: 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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The counit of a reflection is an isomorphism

Statement

Let $R : C \to A$ be a reflector onto a full subcategory, with inclusion $I : A \to C$ , unit $η$ , and counit $ε : R I \Rightarrow 1_{A}$ . Then every component $ε_{A} : R I (A) \to A$ is an isomorphism.

Facts & Assumptions

Given: A reflection $R ⊣ I$ as in Reflective full subcategory and reflector and an object $A \in A$ .

[L1]

A functor $F$ is faithful when every induced $F_{A, B} : C (A, B) \to D (F A, F B)$ is injective, full when every $F_{A, B}$ is surjective, and fully faithful when every $F_{A, B}$ is bijective (Faithful, full, fully faithful, essentially surjective, and split essentially surjective functors).

[L4]

A subcategory $A$ of $C$ is full when $A (A, B) = C (A, B)$ for every pair of its objects (Subcategory and full subcategory).

[L2]

The triangle identities give $I (ε_{A}) \circ η_{I (A)} = 1_{I (A)}$ and $ε_{R (C)} \circ R (η_{C}) = 1_{R (C)}$ (Adjunction by unit, counit, and the triangle identities).

[L3]

A morphism is an isomorphism when it has a two-sided inverse (Isomorphism, groupoid, and connected category).

Proof

technique · direct

1.1

Since $A$ is a full subcategory of $C$ , [L4] gives $A (A, B) = C (A, B) = C (I (A), I (B))$ for all $A, B \in A$ , and the inclusion $I$ acts on those hom-sets as the identity; so every $I_{A, B}$ is bijective and [L1] makes $I$ fully faithful. By its surjectivity there is $δ_{A} : A \to R I (A)$ with $I (δ_{A}) = η_{I (A)}$ , and by its injectivity that $δ_{A}$ is unique. The first triangle identity in [L2] gives $I (ε_{A} \circ δ_{A}) = I (ε_{A}) \circ η_{I (A)} = 1_{I (A)} = I (1_{A})$ , so faithfulness gives $ε_{A} \circ δ_{A} = 1_{A}$ .

L1L2L4

2.1

Naturality of the counit at $δ_{A}$ gives $δ_{A} \circ ε_{A} = ε_{R I (A)} \circ R I (δ_{A})$ . Since $I (δ_{A}) = η_{I (A)}$ , functoriality gives $R I (δ_{A}) = R (η_{I (A)})$ , and the second triangle identity in [L2] makes the right side $1_{R I (A)}$ . Thus $δ_{A}$ is a two-sided inverse of $ε_{A}$ , so [L3] proves the claim.

step 1.1L2L3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 16 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

E. Riehl, Category Theory in Context, lemma 4.5.12 (standard reference, not scraped)