CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

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

1.1L1

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

2.1step 1.1L2L3∎

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].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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