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.

Under local smallness, representable functors give a second proof that right adjoints preserve small limits

Statement

Let $F ⊣ G : D \to C$ be an adjunction between locally small categories. For every small diagram $D : J \to D$ with a limit, the representable-functor calculation identifies $G (\lim D)$ as a limit of $G D$ . Equivalently, the canonical comparison

$G (\lim D) ⟶ \lim (G D)$

is an isomorphism whenever the displayed chosen limits are supplied.

Facts & Assumptions

Given: The locally small adjunction and small diagram in the Statement.

[F1]

A covariantly representable Set-valued functor on a locally small category preserves every small limit that exists (Every covariantly representable functor to Set preserves all existing small limits).

[F2]

A functor preserves a chosen limit exactly when its canonical comparison to the chosen limit of the image diagram is an isomorphism (A functor preserves a chosen limit exactly when its canonical comparison to the chosen target limit is an isomorphism, and dually for colimits).

[L1]

The adjunction gives natural bijections $C (c, G d) ≅ D (F c, d)$ (Under local smallness, transposition gives the natural hom-set bijection, and conversely).

Proof

technique · direct

1.1

For every $c \in C$ , [L1] gives $C (c, G \lim D) ≅ D (F c, \lim D)$ .

2.1

By [F1], the right side is naturally isomorphic to $\lim_{j} D (F c, D_{j})$ , since $D (F c, -)$ is representable and $J$ is small.

step 1.1F1

3.1

Applying [L1] componentwise identifies this limit with $\lim_{j} C (c, G D_{j})$ . The composite is natural in $c$ .

step 2.1L1

4.1

Hence the cone $G (\lim D) \to G D$ represents the cone functor and is limiting. If a chosen limit of $G D$ is also supplied, uniqueness of limits makes the canonical comparison an isomorphism, exactly as stated in [F2].

step 3.1F2∎

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: 34 results over 10 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

Emily Riehl, Category Theory in Context, 2nd ed., Section 4.6 (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Section 6.3 (standard reference, not scraped)