CorollaryStatement: 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.

For a small category, the Yoneda functor preserves and reflects all existing small limits

Statement

For a small category $C$ , its Yoneda embedding $y : C \to [C^{o p}, S e t]$ preserves and reflects every small limit that is defined in $C$ .

Facts & Assumptions

Given: A small category $C$ and a small diagram in it.

[F1]

The Yoneda embedding sends $a$ to $C (-, a)$ (The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding).

[L1]

Covariant representable functors preserve small limits (Every covariantly representable functor to Set preserves all existing small limits).

[L2]

Limits in functor categories are computed pointwise (For small source and index categories, chosen target limits and colimits compute the corresponding functor-category limits and colimits pointwise).

[L3]

Fully faithful functors reflect limits (Fully faithful functors reflect limits and colimits).

[L4]

The Yoneda embedding is fully faithful (The Yoneda functor is fully faithful, and it is a full embedding when its object map is injective).

Proof

technique · pointwise

1.1

At $c \in C$ , evaluation of the Yoneda image of a limiting cone is its image under $C (c, -)$ by [F1]. This is a limiting cone of sets by [L1].

F1L1

2.1

Since every evaluation is limiting, [L2] says the Yoneda-image cone is a limit in the presheaf category. Thus Yoneda preserves the small limit.

L2step 1.1

3.1

By [L4], Yoneda is fully faithful, so [L3] makes it reflect every limit. The smallness of $C$ ensures that the stated presheaf category and embedding are formed under the library convention.

L3L4∎

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: 44 results over 12 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)