TheoremStatement: 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.

General adjoint functor theorem, objectwise initial-object form

Statement

Let $U : A \to C$ , where $A$ is complete and locally small, and suppose $U$ is continuous. Fix $C \in C$ . If $U$ satisfies the solution-set condition at $C$ , then the comma category $(C ↓ U)$ has an initial object. Equivalently, there exists a universal arrow from $C$ to $U$ .

Facts & Assumptions

Given: The functor and hypotheses in the Statement, and one supplied solution set at the fixed object $C$ .

[L1]

A category is complete when every small diagram in it has a limit (Finite, small, and large limits and colimits; complete and cocomplete categories).

[L2]

Local smallness means that each hom-collection is a set (Small, locally small, and large categories).

[L3]

A solution set at $C$ is exactly a jointly weakly initial set in $(C ↓ U)$ (The solution-set condition at an object is exactly a jointly weakly initial set in its comma category).

[L4]

The comma projection strictly creates every projected limit preserved by $U$ (A comma-category projection strictly creates the limits preserved by the functor).

[L5]

A complete locally small category with a supplied jointly weakly initial set has an initial object without class-indexed choice (A complete locally small category with a jointly weakly initial set has an initial object, without class-indexed choice).

[L6]

A functor is continuous when it preserves all small limits (Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors).

Proof

technique · constructive

1.1

The comma category is locally small because each of its hom-collections is a subset of a hom-set in $A$ , which is a set by [L2]. Every small projected diagram has a limit by the completeness of $A$ in the sense of [L1], and $U$ preserves that limit because $U$ is continuous in the sense of [L6], so [L4] creates its limit in the comma category. Hence $(C ↓ U)$ is complete and locally small, and [L3] supplies a jointly weakly initial set.

L1L2L3L4L6construct

2.1

Apply [L5] to obtain an initial object of $(C ↓ U)$ . Its construction uses only the supplied solution set for this fixed $C$ , so it performs no simultaneous selection over all objects of $C$ .

step 1.1L5discharge-construct∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 35 results over 11 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, theorem 4.7.3 (standard reference, not scraped)
T. Leinster, Basic Category Theory, theorem 6.3.10 (standard reference, not scraped)