CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16‡ rests on unproved material (inherited)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

With the objectwise SAFT universal arrows supplied, a continuous Set-valued functor from a chosen-well-powered SAFT category is representable

Statement

Let $C$ be complete and locally small, with a supplied small coseparating set and a supplied well-powering. Let $F : C \to S e t$ be continuous. If a supplied family of the objectwise SAFT universal arrows is given, then $F$ is covariantly representable.

Facts & Assumptions

Given: The category, functor, and supplied SAFT data in the Statement.

[L1]

Under the supplied-well-powering branch, objectwise SAFT produces initial objects in the comma categories of a continuous functor, and supplied initial objects assemble into a left adjoint (Special adjoint functor theorem, objectwise form with explicit intersection smallness or preservation data, Special adjoint functor theorem, data-supplied functor form).

[L2]

A covariant Set-valued functor is representable when it is naturally isomorphic to $C (R, -)$ for some $R$ (Presheaves, covariantly and contravariantly representable functors, and representations).

[L3]

For locally small $C$ and $D$ , an adjunction $F ⊣ G$ determines bijections $Φ_{c, d} : D (F c, d) \to C (c, G d)$ , $Φ_{c, d} (u) = G (u) \circ η_{c}$ , natural in $c$ and $d$ (Under local smallness, transposition gives the natural hom-set bijection, and conversely).

Proof

technique · direct

1.1

By [L1], the supplied universal arrows assemble into a left adjoint $L : S e t \to C$ to $F$ .

2.1

Let $1$ be a singleton set. The adjunction bijection in [L3] gives $C (L (1), C) ≅ S e t (1, F (C)) ≅ F (C)$ , naturally in $C$ . Hence $F$ is represented by $L (1)$ in the sense of [L2].

step 1.1L2L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 48 results over 18 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

S. Mac Lane, Categories for the Working Mathematician, section V.8 (standard reference, not scraped)