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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A full subcategory is reflectively structured exactly when universal arrows are supplied at every ambient object

Statement

Let $A$ be a full subcategory of $C$ , with inclusion $I : A \to C$ . The following supplied data are equivalent:

a reflector $R : C \to A$ and an adjunction $R ⊣ I$ (Reflective full subcategory and reflector);
for every object $C \in C$ , a specified universal arrow $(R_{C}, η_{C})$ from $C$ to $I$ (Universal arrows from an object to a functor and from a functor to an object).

Under this equivalence the specified universal arrows are the components of the reflection unit. Merely asserting their existence, without supplying them object by object, does not supply the functor data in item 1.

Facts & Assumptions

Given: A full inclusion $I : A \to C$ .

[L1]

A reflection consists of a left adjoint $R$ to $I$ , with unit components $η_{C} : C \to I R (C)$ (Reflective full subcategory and reflector).

[L2]

A universal arrow from $C$ to $I$ is a pair $(A, η : C \to I (A))$ such that every $f : C \to I (B)$ factors uniquely as $f = I (h) \circ η$ (Universal arrows from an object to a functor and from a functor to an object).

[L3]

A left adjoint to $I$ is supplied exactly by choosing an initial object, equivalently a universal arrow, in every comma category $(C ↓ I)$ ; the supplied objects determine the functor on morphisms and the adjunction uniquely (A left adjoint exists exactly when chosen initial objects are supplied in every comma category).

Proof

technique · direct

1.1

Suppose item 1 is supplied. By [L3] a supplied left adjoint to $I$ corresponds to an initial object, equivalently a universal arrow, of each comma category $(C ↓ I)$ , and the unit components are exactly those arrows. Hence for each $C$ the unit component $η_{C} : C \to I R (C)$ has the universal factorisation property of [L2], and $(R (C), η_{C})$ is the required specified universal arrow.

L1L2L3

2.1

Conversely, suppose item 2 is supplied. By [L3] the specified initial objects $(R_{C}, η_{C})$ of $(C ↓ I)$ determine a functor $R : C \to A$ and an adjunction $R ⊣ I$ whose unit is $(η_{C})$ ; hence $A$ is reflective by [L1]. No selection beyond the supplied family is made.

L1L2L3step 1.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 21 results over 12 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 4.5.11 (standard reference, not scraped)
T. Leinster, Basic Category Theory, section 6.3 (standard reference, not scraped)