# Yoneda lemma

The Yoneda lemma is a theorem of category theory stating that, for a category \( \mathcal{C} \), an object \( C \), and a set-valued functor \( F \), the natural transformations from the hom-functor \( \mathcal{C}(C,-) \) to \( F \) are in natural bijection with the set \( F(C) \). The bijection sends a natural transformation \( \alpha \) to the value \( \alpha_C(\mathrm{id}_C) \) of its component at \( C \) on the identity morphism.<sup>[1](https://www.math.uni-hamburg.de/home/richter/bookdraft.pdf)</sup><sup> • </sup><sup>[2](https://categorytheory.gitlab.io/yoneda.html)</sup>

| Key fact | Detail |
|---|---|
| Statement | For \( F : \mathcal{C}^{\mathrm{op}} \to \mathbf{Set} \) and an object \( B \), \( \mathrm{Nat}(Y_B, F) \cong FB \), naturally in \( B \) and \( F \), where \( Y_B = \mathcal{C}(-,B) \)<sup>[2](https://categorytheory.gitlab.io/yoneda.html)</sup> |
| Evaluation map | A transformation \( \alpha : \mathcal{C}(C,-) \Rightarrow F \) is sent to \( \alpha_C(\mathrm{id}_C) \in F(C) \); the inverse reconstructs \( \alpha \) from that single element<sup>[3](https://sites.math.washington.edu/~jarod/moduli.pdf)</sup> |
| Yoneda embedding | \( h : \mathcal{C} \to \hat{\mathcal{C}} \), \( X \mapsto \mathrm{Hom}_{\mathcal{C}}(-,X) \), is fully faithful<sup>[4](https://math.uchicago.edu/~amathew/232b.pdf)</sup> |
| Consequence for representables | A representable functor is representable by a unique object up to isomorphism; two objects are isomorphic exactly when their representables are<sup>[3](https://sites.math.washington.edu/~jarod/moduli.pdf)</sup><sup> • </sup><sup>[5](https://notes.amittai.studio/category-theory/representables-yoneda/yoneda-consequences)</sup> |
| Size hypothesis | Typically applied to locally small categories; for a locally small \( \mathcal{A} \), \( \mathrm{Nat}(H_A, X) \) need not a priori be a set, and the lemma shows it is one, in bijection with \( X(A) \)<sup>[6](https://math-cs-compass.com/Mathematics/Discrete/yoneda_lemma.html)</sup> |
| Origin | Nobuo Yoneda, "On the homology theory of modules", *J. Fac. Sci. Univ. Tokyo Sect. I* 7 (1954), pp. 193–227<sup>[7](https://stacks.math.columbia.edu/bibliography/Yoneda-homology)</sup> |

## Statement and mechanism

In the covariant form, for each object \( C \) of \( \mathcal{C} \) there is a bijection \( Y(F,C) \) between the set \( \mathrm{Nat}(\mathcal{C}(C,-), F) \) of natural transformations from \( \mathcal{C}(C,-) \) to \( F \) and the set \( F(C) \); these bijections are themselves the components of a natural transformation as \( C \) varies.<sup>[1](https://www.math.uni-hamburg.de/home/richter/bookdraft.pdf)</sup> In the contravariant (presheaf) form used most often, \( F : \mathcal{C}^{\mathrm{op}} \to \mathbf{Set} \) and \( \mathrm{Nat}(Y_B, F) \cong FB \), with the isomorphism natural in both \( B \) and \( F \).<sup>[2](https://categorytheory.gitlab.io/yoneda.html)</sup>

The mechanism is evaluation at the identity. Given \( \alpha \in \mathrm{Nat}(Y_B, F) \), one forms \( x_\alpha = \alpha_B(\mathrm{id}_B) \in FB \).<sup>[2](https://categorytheory.gitlab.io/yoneda.html)</sup> In the Washington moduli notes the map is written \( \mathrm{Mor}(h_X, G) \to G(X) \), \( \alpha \mapsto \alpha_X(\mathrm{id}_X) \), where \( h_X(S) = \mathrm{Mor}(S,X) \); it is bijective and functorial in both \( X \) and \( G \).<sup>[3](https://sites.math.washington.edu/~jarod/moduli.pdf)</sup> The inverse sends an element \( u \in F(C) \) to the transformation whose component at \( X \) carries a morphism \( f : X \to C \) to \( F(f)(u) \); naturality of \( \alpha \) is exactly what makes this well defined.

A corollary concerns size. For a merely locally small category \( \mathscr{A} \), the collection of natural transformations between two presheaves has one component per object and need not, in general, be a set. The lemma includes the assertion that when the source is a representable \( H_A \), this collection is a set, in bijection with \( X(A) \), with no appeal to a universe or to global smallness.<sup>[6](https://math-cs-compass.com/Mathematics/Discrete/yoneda_lemma.html)</sup>

## The Yoneda embedding

Applying the lemma with \( F = h_{X} \) shows that the Yoneda embedding
\[ h : \mathcal{C} \to \hat{\mathcal{C}}, \qquad h_X(Y) = \mathrm{Hom}_{\mathcal{C}}(Y,X) \]
into the category \( \hat{\mathcal{C}} \) of contravariant set-valued functors is fully faithful.<sup>[4](https://math.uchicago.edu/~amathew/232b.pdf)</sup> Faithfulness follows because a natural transformation is determined by where it sends the identity: if \( h^{\mathcal{C}}(f) = h^{\mathcal{C}}(g) \), then \( f = f \circ \mathrm{id}_A = h^{\mathcal{C}}(f)_A(\mathrm{id}_A) = h^{\mathcal{C}}(g)_A(\mathrm{id}_A) = g \).<sup>[8](https://www.math.ucla.edu/~sharifi/notes/homalg-ch01.html)</sup> Fullness says every natural transformation \( h^A \Rightarrow h^B \) arises from a morphism \( A \to B \), namely \( \eta_A(\mathrm{id}_A) \).<sup>[8](https://www.math.ucla.edu/~sharifi/notes/homalg-ch01.html)</sup>

Two consequences follow directly. An object is determined up to isomorphism by its representable functor, so two objects are isomorphic exactly when their representables are.<sup>[5](https://notes.amittai.studio/category-theory/representables-yoneda/yoneda-consequences)</sup> And a representation of a presheaf is the same thing as a universal element.<sup>[5](https://notes.amittai.studio/category-theory/representables-yoneda/yoneda-consequences)</sup> The embedding also lets any category \( \mathcal{C} \) sit inside its category of presheaves.<sup>[2](https://categorytheory.gitlab.io/yoneda.html)</sup>

## Representable functors

A functor \( F : \mathcal{C} \to \mathbf{Set} \) is representable if it is naturally isomorphic to a hom-functor \( \mathcal{C}(C,-) \); the pair \( (C, \eta) \) is called a representation.<sup>[1](https://www.math.uni-hamburg.de/home/richter/bookdraft.pdf)</sup><sup> • </sup><sup>[9](https://math.uchicago.edu/~may/REU2022/REUPapers/Stern.pdf)</sup> By the Yoneda lemma, a representing object is unique up to isomorphism: in the scheme-theoretic setting, if a functor is representable, it is representable by a unique scheme.<sup>[3](https://sites.math.washington.edu/~jarod/moduli.pdf)</sup>

Standard examples show the range of the notion:

- The forgetful functor \( U : \mathbf{Group} \to \mathbf{Set} \) is representable by the group \( \mathbb{Z} \); for each group element \( g \) one uses the unique homomorphism \( \mathbb{Z} \to G \) carrying 1 to \( g \).<sup>[9](https://math.uchicago.edu/~may/REU2022/REUPapers/Stern.pdf)</sup>
- The functor sending a topological space \( X \) to its set of open subsets is representable by the Sierpinski space, since continuous maps into it are exactly the characteristic functions of open subsets.<sup>[10](https://danielnaylor.uk/notes/III/Michaelmas/CT/HTML/CTse2.html)</sup>
- For a group \( G \) regarded as a one-object category, the unique representable functor is the Cayley representation, \( G \) acting on itself by multiplication.<sup>[10](https://danielnaylor.uk/notes/III/Michaelmas/CT/HTML/CTse2.html)</sup> This illustrates the general point, made in the Barr–Wells lecture notes, that representable functors generalize the regular representation and the Yoneda embedding generalizes Cayley's theorem.<sup>[11](https://www.math.upatras.gr/~cdrossos/Docs/B-W-LectureNotes.pdf)</sup>
- Categorical products and equalizers can be defined as representing objects of suitable functors, which recovers their universal properties.<sup>[10](https://danielnaylor.uk/notes/III/Michaelmas/CT/HTML/CTse2.html)</sup>

## Uses in algebraic geometry

Grothendieck's functor-of-points approach studies a scheme \( X \) through the functor \( h_X = \mathrm{Hom}_{\mathbf{Sch}}(-,X) \), assigning to each test scheme \( T \) the set of \( T \)-valued points of \( X \). The Yoneda lemma guarantees that this functor determines \( X \) completely up to isomorphism, since \( h_{(-)} : \mathbf{Sch} \to \mathrm{Fun}(\mathbf{Sch}^{\mathrm{op}}, \mathbf{Set}) \) is fully faithful.<sup>[12](https://math-jh.com/en/math/scheme_theory/functor_of_points)</sup> The same lemma underlies the definition of group schemes: a group object is a group functor \( G : \mathcal{C}^{\mathrm{op}} \to \mathbf{Grp} \) isomorphic to some \( h_X \), and conversely, using the Yoneda lemma, group data on \( h_X \) yields multiplication, inversion, and identity maps \( m : X \times X \to X \), \( s : X \to X \), \( e : * \to X \) satisfying the group axioms.<sup>[4](https://math.uchicago.edu/~amathew/232b.pdf)</sup>

Moduli theory pushes this further. A moduli problem is a functor \( h : \mathbf{Sch}^{\mathrm{op}} \to \mathbf{Sets} \) sending a scheme \( S \) to the set of families parametrized by \( S \); if \( h \) is representable, then by the Yoneda lemma there exists a unique \( X \) with \( h = h_X \).<sup>[13](https://www.ma.imperial.ac.uk/~rpwt/LMSnotes.pdf)</sup> When no scheme represents the functor, the problem may still be representable by a stack, which motivates 2-categorical versions of the Yoneda lemma.<sup>[14](https://arxiv.org/abs/2202.06628)</sup>

## Generalizations

The lemma extends along several axes. In enriched category theory, a \( V \)-enriched functor \( F : C \to V \) is representable when there is an object \( c \) and a \( V \)-enriched natural transformation \( \eta : F \to C(c,-) \), and the lemma generalizes to this setting.<sup>[15](https://ncatlab.org/nlab/show/representable%2Bfunctor)</sup> In \( \infty \)-category theory, the Yoneda embedding \( \mathscr{X} \to \mathrm{Fun}(\mathscr{X}^{\mathrm{op}}, \mathcal{S}) \), \( x \mapsto \mathscr{X}(-,x) \), is fully faithful for a locally small \( \infty \)-category, and the lemma has been formalized constructively: in homotopy type theory it gives, for a pre-∞-category \( A \), an equivalence between the type \( (z : A) \to \mathrm{hom}\,A\,a\,z \to C z \) of natural transformations out of the representable family and the type \( C a \), with evaluation at the identity as one direction.<sup>[16](https://arxiv.org/abs/2607.14016)</sup><sup> • </sup><sup>[17](https://emilyriehl.github.io/yoneda/CPP-2024/simplicial-hott/09-yoneda.rzk/)</sup> There are also internal versions: in a finitely complete, locally Cartesian closed \( \infty \)-category with a universe classifying the diagonal of \( X \), the internal Yoneda map \( \mathscr{Y}_X : X \to \mathscr{U}^X \) is a monomorphism, proved using only finite limits, dependent products, and universes.<sup>[16](https://arxiv.org/abs/2607.14016)</sup> Double categories admit a Yoneda lemma and representation theorem for lax double presheaves as well.<sup>[18](https://arxiv.org/abs/2402.10640)</sup>

The lemma also functions as an engine for other theorems. A 2023 note shows how two fundamental results of topos theory, including the classification of site morphisms \( \mathcal{C} \to \mathcal{Z} \) into any topos by the canonical functor into sheaves, follow by repeated application of the Yoneda lemma together with the formalism of natural transformations.<sup>[19](https://arxiv.org/abs/2312.04716)</sup>

## References

1. [Birgit Richter, *Category Theory* book draft, §2.2 The Yoneda Lemma, Universität Hamburg](https://www.math.uni-hamburg.de/home/richter/bookdraft.pdf)
2. [Category Theory: a concise course, §8 Yoneda's Lemma](https://categorytheory.gitlab.io/yoneda.html)
3. [Jarod Alper (moduli notes), §1.3.3 Yoneda's lemma and representable functors, University of Washington](https://sites.math.washington.edu/~jarod/moduli.pdf)
4. [Arun Mathew, Algebraic geometry notes, Lecture 8: Group schemes, University of Chicago](https://math.uchicago.edu/~amathew/232b.pdf)
5. [The Yoneda Embedding and Its Uses, Amittai study notes](https://notes.amittai.studio/category-theory/representables-yoneda/yoneda-consequences)
6. [Yoneda Lemma, MATH-CS COMPASS](https://math-cs-compass.com/Mathematics/Discrete/yoneda_lemma.html)
7. [Bibliography entry Yoneda-homology, The Stacks Project](https://stacks.math.columbia.edu/bibliography/Yoneda-homology)
8. [Romyar Sharifi, Homological Algebra, Chapter 1: Category Theory, UCLA](https://www.math.ucla.edu/~sharifi/notes/homalg-ch01.html)
9. [Noah Stern, Introduction to Category Theory and the Yoneda Lemma, REU papers, University of Chicago](https://math.uchicago.edu/~may/REU2022/REUPapers/Stern.pdf)
10. [The Yoneda Lemma, Category Theory, Cambridge Part III notes by Daniel Naylor](https://danielnaylor.uk/notes/III/Michaelmas/CT/HTML/CTse2.html)
11. [Barr & Wells, Category Theory lecture notes, §5.7, University of Patras mirror](https://www.math.upatras.gr/~cdrossos/Docs/B-W-LectureNotes.pdf)
12. [Functor of Points, Blackbox mathematics notes](https://math-jh.com/en/math/scheme_theory/functor_of_points)
13. [LMS notes on representability and moduli problems, Imperial College London](https://www.ma.imperial.ac.uk/~rpwt/LMSnotes.pdf)
14. [Notes on Moduli theory, Stacks and 2-Yoneda's Lemma, arXiv:2202.06628](https://arxiv.org/abs/2202.06628)
15. [representable functor, nLab](https://ncatlab.org/nlab/show/representable%2Bfunctor)
16. [The internal Yoneda Lemma for locally Cartesian closed ∞-categories, arXiv:2607.14016](https://arxiv.org/abs/2607.14016)
17. [The Yoneda Lemma, Yoneda for ∞-categories (Riehl et al., formalization in Rzk)](https://emilyriehl.github.io/yoneda/CPP-2024/simplicial-hott/09-yoneda.rzk/)
18. [Yoneda lemma and representation theorem for double categories, arXiv:2402.10640](https://arxiv.org/abs/2402.10640)
19. [From Yoneda to Topoi morphisms, arXiv:2312.04716](https://arxiv.org/abs/2312.04716)

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