Given an object X of C, we define a contravariant functor hX : C → Sets from C to the category of sets as follows

1. Yoneda Lemma

Let F, G : C → D be functors between two categories C and D. A natural transformation η : F → G is a family of morphisms ηX : F (X) → G(X) parametrized by objects of C such that for every morphism f : X → Y, the following diagram commutes:

(1.1)

F (X) −−−−→ G(X)^ηX

F (f )





y ^{G(f )}



 y F (Y ) −−−−→ G(Y ).^ηY

Similarly, we can define natural transformations between contravariant functors: in the above definition, we replace f : X → Y by a morphism f : Y → X.

Let C be a category. Given an object X of C, we define a contravariant functor hX : C → Sets

from C to the category of sets as follows. For each object Y, define h_X(Y ) = HomC(Y, X) and for each f : Z → Y, we define hX(f ) : HomC(Y, X) → HomC(Z, X) by hX(f )(u) = u ◦ f, where u : Y → X is any morphism in C.

Definition 1.1. A contravariant functor F : C → Sets is representable if there exists an object X of C so that F ∼= hX. The object X is called a representative of F.

We will show that such X is unique up to an isomorphism.

A functor F : C → C⁰ is faithful if for any objects X, X⁰ of C, the map HomC(X, X⁰) → HomC⁰(F (X), F (X⁰))

is injective and full if it is surjective. A functor is fully faithful if the above map is a bijection.

Let C^∨ be the category of all contravariant functors from C to the category of sets. Then we obtain a functor

h : C → C^∨, X 7→ h_X. Proposition 1.1. Let h : C → C^∨ be as above.

(1) For any object X of C, we have a bijection

HomC^∨(hX, F ) ⇐⇒ F (X).

(2) The functor h is fully faithful.

Proof. Let φ : h_X → F be a natural transformation. Then η_X : h_X(X) → F (X) is a map of sets. Since hX(X) = HomC(X, X), 1X belongs to hX(X). Therefore ηX(1X) gives an element of F (X). Define a map

α : HomC^∨(hX, F ) → F (X) by η 7→ η_X(1_X).

Let s ∈ F (X). For any object Y of C, and any morphism ψ ∈ h_X(Y ) = HomC(Y, X), we have F (ψ) ∈ HomSets(F (X), F (Y )). Since s ∈ F (X), we obtain F (ψ)(s) ∈ F (Y ). This observation allows us to define a map

β : F (X) → HomC^∨(hX, F )

as follows. For each s ∈ F (X), β(s) defines a natural transformation from h_X to F by βY(s) : HomC(Y, X) → F (Y ), ψ 7→ F (ψ)(s)

1

2

for each object Y of C. Check that α, β are inverse to each other. To do this, let us consider the following diagram: given η ∈ HomC^∨(h_X, F ), and φ : Y → X,

(1.2)

HomC(X, X) −−−−→ F (X)^ηX

hX(φ)





y ^{F (φ)}



 y HomC(Y, X) −−−−→ F (Y ).^ηY

Using this diagram, we find η_Y ◦ h_X(φ)(1_X) = F (φ) ◦ η_X(1_X). Since h_X(φ)(1_X) = φ, we find ηY(φ) = F (φ)(ηX(1X)) = βY(α(η))(φ), ∀φ. This shows that η_Y = βY(α(η)) for all objects Y of C, and hence η = (β ◦ α)(η) for all η ∈ HomC^∨(h_X, F ). This shows that

β ◦ α = 1_Hom

C∨(hX,F ). Now, let s ∈ F (X), then

(α ◦ β)(s) = α(β(s)) = βX(s)(1X) = F (1X)(s) = 1_{F (X)}(s) = s.

This shows that

α ◦ β = 1_{F (X)}.

We find that α, β are inverse to each other. Therefore α is a bijection.

To prove (2), let us choose F = h_X⁰ for any object X⁰ in C⁰. (1) implies the bijection of the set

HomC(X, X⁰) ⇐⇒ HomC^∨(hX, hX⁰).

By definition, h : C → C^∨ sending X → h_X is fully faithful.  Corollary 1.1. Let C be a category. Suppose that X and X⁰ are objects of C so that h_X ∼= hX⁰. Then X and X⁰ are isomorphic.

Proof. Suppose that h_X and h_X⁰ are isomorphic. Then there is an isomorphism η : h_X → h_X⁰

whose inverse is denoted by η⁻¹ : hX⁰ → h_X. Denote the map α_X,X⁰ = HomC^∨(hX, h_X⁰) → HomC(X, X⁰).

Then by definition,

α_X,X(η⁻¹◦ η) = α_X⁰_,X(η⁻¹) ◦ α_X,X⁰(η), α_X⁰_,X⁰(η ◦ η⁻¹) = α_X,X⁰(η) ◦ α_X⁰_,X(η⁻¹).

By αX,X(1hX) = 1X and αX⁰,X⁰(1_h⁰

X) = 1X⁰, we find

α_X⁰_,X(η⁻¹) ◦ α_X,X⁰(η) = 1_X, α_X,X⁰(η) ◦ α_X⁰_,X(η⁻¹) = 1_X⁰.

This shows that α_X,X⁰(η) : X → X⁰ and α_X⁰_,X(η⁻¹) : X⁰ → X are inverse to each other.

Therefore X and X⁰ are isomorphic. 

This Corollary shows that any two representatives of a representable functor are isomor- phic.