TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13‡ 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.

Every covariantly representable functor to Set preserves all existing small limits

Statement

Let $C$ be locally small and let $R : C \to S e t$ be covariantly representable. For every small diagram in $C$ whose limit exists, its image under $R$ is a limit in $S e t$ .

Facts & Assumptions

Given: A small $D : J \to C$ , a limit $(L, λ)$ , and a representation $R ≅ C (X, -)$ .

[F1]

Covariant and contravariant hom-assignments have their stated actions on morphisms and are functors (The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category, The assignments $C (a, -)$ and $C (-, a)$ are functors to $S e t$ ).

[F2]

A covariantly representable functor is naturally isomorphic to $C (X, -)$ (Presheaves, covariantly and contravariantly representable functors, and representations).

[F3]

A limit represents compatible cones by unique arrows (Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties).

[L1]

Every small set-valued diagram has a compatible-tuple limit (Set has all small limits, realized as compatible tuples in a set-indexed product).

[L2]

Yoneda's bijection and its inverse are natural in both variables (The Yoneda bijection $Nat (C (a, -), F) ≅ F (a)$ is natural in both $a$ and $F$ ).

Proof

technique · universal property

1.1

For the hom-functor, define $Φ : C (X, L) \to \prod_{j} C (X, D (j))$ by $Φ (f)_{j} = λ_{j} f$ . The cone equations put its image in the compatible subset that [L1] identifies as $\lim_{j} C (X, D (j))$ .

F1F3L1

1.2

Conversely, a compatible family $(f_{j} : X \to D (j))_{j}$ is a cone over $D$ . By [F3] there is a unique $f : X \to L$ with $λ_{j} f = f_{j}$ . This defines an inverse $Ψ$ to $Φ$ .

2.1

The equations in steps 1.1 and 1.2 give $Ψ Φ (f) = f$ by limit uniqueness and $Φ Ψ (f_{j}) = (f_{j})$ coordinatewise. Thus the image cone under $C (X, -)$ is a Set-limit.

F3step 1.1step 1.2

3.1

The natural isomorphism in [F2] transports this limiting cone to the image under $R$ ; its compatibility follows from naturality, equivalently from [L2]. Hence $R$ preserves the limit. Smallness is needed so the limit in [L1] is a set.

F2L1L2step 2.1∎

Depends on

Used by

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

E. Riehl, Category Theory in Context, Theorem 3.5.5 (standard reference, not scraped)