Mathlib definition · Existing formal mathematics
ProofAtlas

Existing in Mathlib · Category theory

Yoneda Lemma

To specify a natural transformation from the presheaf of arrows into X to a presheaf F, it is enough to choose one element of F at X. Evaluating the transformation on the identity of X retrieves that element, and functorial transport reconstructs every component.

Exact definition

Exact Mathlib definition

def CategoryTheory.yonedaEquiv {C : Type u₁} [Category.{v₁} C] {X : C} {F : Cᵒᵖ ⥤ Type v₁} : (CategoryTheory.yoneda.obj X ⟶ F) ≃ F.obj (Opposite.op X)

The theorem at a glance

Yoneda lemma at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

Evaluation at identity_X and functorial reconstruction give the Yoneda equivalence. Explanatory diagram.
Detailed visual description

The poster makes the type equivalence—not a proposition—the central object. One side displays a natural transformation with components over many Y; the other displays a single ξ in F(X). Forward evaluation and inverse transport are shown as genuine inverse directions.

Statement structure

Statement and scope

Statement map for Yoneda LemmaNatural transformations from the representable presheaf Hom(−,X) to F are equivalent to elements of F(X). Claim boundary: This target indexes Mathlib's definition of the type-level Yoneda equivalence for presheaves valued in Type at the category's morphism universe. It is a constructed Equiv, not a proposition-valued theorem. The selected endpoint does not by itself state the Yoneda embedding's full faithfulness, although it is central to that theory. The pinned upstream declaration is CategoryTheory.yonedaEquiv. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingNatural transformationsfrom the representablepresheaf Hom(−,X) to Fare equivalent toelements of F(X).Claim boundaryThis target indexesMathlib's definition ofthe type-level Yonedaequivalence forpresheaves valued inType at the category'smorphism universe. It isa constructed Equiv, nota proposition-valuedtheorem. The selectedendpoint does not byitself state the Yonedaembedding's fullfaithfulness, althoughit is central to thattheory.Pinned declarationmathlib ·CategoryTheory.yonedaEquivStatement map for Yoneda LemmaNatural transformations from the representable presheaf Hom(−,X) to F are equivalent to elements of F(X). Claim boundary: This target indexes Mathlib's definition of the type-level Yoneda equivalence for presheaves valued in Type at the category's morphism universe. It is a constructed Equiv, not a proposition-valued theorem. The selected endpoint does not by itself state the Yoneda embedding's full faithfulness, although it is central to that theory. The pinned upstream declaration is CategoryTheory.yonedaEquiv. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingNatural transformationsfrom the representablepresheaf Hom(−,X) to Fare equivalent toelements of F(X).Claim boundaryThis target indexesMathlib's definition ofthe type-level Yonedaequivalence forpresheaves valued inType at the category'smorphism universe. It isa constructed Equiv, nota proposition-valuedtheorem. The selectedendpoint does not byitself state the Yonedaembedding's fullfaithfulness, althoughit is central to thattheory.Pinned declarationmathlib ·CategoryTheory.yonedaEquiv

Read the exact Mathlib declaration

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

A natural transformation out of Hom(−,X) is completely encoded by its value on the identity of X. Explanatory diagram.
Detailed visual description

Several objects Y send morphisms into a central X, forming the representable presheaf. On the opposite side, functorial images radiate from one distinguished element at F(X); a two-way gold seam pairs this element with the complete natural transformation.

Why it matters

A mathematical landmark

The Yoneda lemma says that an object is understood through all arrows into it. It is the organizing correspondence behind representability, universal properties, and the Yoneda embedding.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally
Locally reproducedExact upstream declaration replayed
Reviewed pageCurrent public presentation reviewed
Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This target indexes Mathlib's definition of the type-level Yoneda equivalence for presheaves valued in Type at the category's morphism universe. It is a constructed Equiv, not a proposition-valued theorem. The selected endpoint does not by itself state the Yoneda embedding's full faithfulness, although it is central to that theory.

Source and local evidence

Where the theorem comes from

Existing declaration
CategoryTheory.yonedaEquiv in mathlib
Relationship
The checked artifact is the upstream declaration itself
Local Lean evidence
artifact.library.mathlib.yoneda-lemma.v001
Source
Open the pinned upstream reference