CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-05

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.

The enriched Yoneda assignment is fully faithful

Statement

Whenever the enriched Yoneda assignment is formed, it is fully faithful: for objects $A, B$ of a $V$ -category $C$ , the hom-object between the representables $C (-, A)$ and $C (-, B)$ is naturally isomorphic to $C (A, B)$ .

Facts & Assumptions

Given: A $V$ -category $C$ and objects $A, B$ .

[L1]

The representable enriched functor at an object is $C (-, B)$ or dually $C (B, -)$ (Representable enriched functor).

[L2]

The strong enriched Yoneda lemma identifies $F B$ with the end $\int_{X} [C (B, X), F X]$ (Strong enriched Yoneda lemma as a particular end).

Proof

technique · direct

1.1

By [L1], the hom-object from the contravariant representable $C (-, A)$ to the contravariant representable $C (-, B)$ is the end $\int_{X} [C (X, A), C (X, B)] .$

L1given

2.1

Apply [L2] to the $V$ -category $C^{o p}$ , the object $A$ , and the $V$ -functor $C (-, B) : C^{o p} \to V$ . Since $C^{o p} (A, X) = C (X, A)$ , it identifies the end of step 1.1 with $(C (-, B)) (A) = C (A, B)$ .

L1L2step 1.1

3.1

Therefore each hom-object map of the enriched Yoneda assignment is an isomorphism, which is exactly enriched full faithfulness.

step 2.1∎

Depends on

Used by

FALSE: the strong enriched Yoneda lemma for a large category constructs the whole enriched functor category False statement

Dependency tree · two levels

7 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

G. M. Kelly, Basic Concepts of Enriched Category Theory, equation (2.33) (standard reference, not scraped)
Emily Riehl, Categorical Homotopy Theory, Section 7.3 (standard reference, not scraped)