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

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

Statement

Let $C$ be locally small. For all objects $a, b$ , the Yoneda assignment induces a bijection

$C (a, b) \overset{≅}{\to} Nat (C (-, a), C (-, b)), f ⟼ y (f),$

where $y (f)_{c} (g) = f \circ g$ . Thus, when $C$ is small and $y$ is the functor of The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding, it is fully faithful. If its object map is also injective, it is a full embedding in the library's stronger sense.

Facts & Assumptions

Given: A locally small category $C$ and objects $a, b$ .

[L1]

Contravariant Yoneda gives $Nat (C (-, a), P) ≅ P (a)$ by evaluation, with inverse $x \mapsto (f \mapsto P (f) (x))$ (For a presheaf $P$ , $Nat (C (-, a), P) ≅ P (a)$ naturally in $a$ and $P$ ).

[F1]

The Yoneda assignment sends $f : a \to b$ to postcomposition $g \mapsto f \circ g$ (The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding).

[F2]

A functor is fully faithful exactly when each induced hom-map is bijective (Faithful, full, fully faithful, essentially surjective, and split essentially surjective functors).

[F3]

An embedding is faithful and injective on objects, and a full embedding is additionally full (Embedding and full embedding of categories).

Proof

technique · direct

1.1

Apply [L1] to $P = C (-, b)$ . Evaluation sends $α : C (-, a) \Rightarrow C (-, b)$ to $α_{a} (1_{a}) \in C (a, b)$ , and its inverse sends $f : a \to b$ to $g : c \to a \mapsto f \circ g$ , which is exactly $y (f)$ by [F1].

L1F1

2.1

Step 1.1 makes every Yoneda hom-map bijective, so [F2] gives full faithfulness whenever the small-source Yoneda functor is formed; the same bijections hold objectwise for every locally small $C$ .

step 1.1F2

3.1

If the object map of $y$ is injective, step 2.1 gives both fullness and faithfulness, so [F3] makes $y$ a full embedding.

step 2.1F3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 28 results over 11 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, Corollary 2.2.8 (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Corollary 4.3.7 (standard reference, not scraped)