Yoneda lemma
Formal statement

Yoneda's lemma concerns functors from a fixed category C {\displaystyle {\mathcal {C}}} to the category of sets, S e t {\displaystyle \mathbf {Set} }. If C {\displaystyle {\mathcal {C}}} is a locally small category (i.e. the hom-sets are actual sets and not proper classes), then each object A {\displaystyle A} of C {\displaystyle {\mathcal {C}}} gives rise to a natural functor to S e t {\displaystyle \mathbf {Set} } called a hom-functor. This functor is denoted:

The (covariant) hom-functor h A {\displaystyle h_{A}} sends X ∈ C {\displaystyle X\in {\mathcal {C}}} to the set of morphisms H o m (A, X) {\displaystyle \mathrm {Hom} (A,X)} and sends a morphism f: X → Y {\displaystyle f\colon X\to Y} (where Y ∈ C {\displaystyle Y\in {\mathcal {C}}}) to the morphism f ∘ − {\displaystyle f\circ -} (composition with f {\displaystyle f} on the left) that sends a morphism g {\displaystyle g} in H o m (A, X) {\displaystyle \mathrm {Hom} (A,X)} to the morphism f ∘ g {\displaystyle f\circ g} in H o m (A, Y) {\displaystyle \mathrm {Hom} (A,Y)}. That is,

Yoneda's lemma says that:

Lemma (Yoneda) — Let F {\displaystyle F} be a functor from a locally small category C {\displaystyle {\mathcal {C}}} to S e t {\displaystyle \mathbf {Set} }. Then for each object A {\displaystyle A} of C {\displaystyle {\mathcal {C}}}, the natural transformations N a t (h A, F) ≡ H o m (H o m (A, −), F) {\displaystyle \mathrm {Nat} (h_{A},F)\equiv \mathrm {Hom} (\mathrm {Hom} (A,-),F)} from h A {\displaystyle h_{A}} to F {\displaystyle F} are in one-to-one correspondence with the elements of F (A) {\displaystyle F(A)}. That is,

Moreover, this isomorphism is natural in A {\displaystyle A} and F {\displaystyle F} when both sides are regarded as functors from C × S e t C {\displaystyle {\mathcal {C}}\times \mathbf {Set} ^{\mathcal {C}}} to S e t {\displaystyle \mathbf {Set} }.

Here the notation S e t C {\displaystyle \mathbf {Set} ^{\mathcal {C}}} denotes the category of functors from C {\displaystyle {\mathcal {C}}} to S e t {\displaystyle \mathbf {Set} }.

Given a natural transformation Φ {\displaystyle \Phi } from h A {\displaystyle h_{A}} to F {\displaystyle F}, the corresponding element of F (A) {\displaystyle F(A)} is u = Φ A (i d A) {\displaystyle u=\Phi _{A}(\mathrm {id} _{A})}; and given an element u {\displaystyle u} of F (A) {\displaystyle F(A)}, the corresponding natural transformation is given by Φ X (f) = F (f) (u) {\displaystyle \Phi _{X}(f)=F(f)(u)} which assigns to a morphism f: A → X {\displaystyle f\colon A\to X} a value of F (X) {\displaystyle F(X)}.
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