Let C be a locally small category (i.e. a category for which Hom-classes are actually sets and not proper classes). For all objects A in C we define a covariant functor
Hom(A,–) : C → Set
to the category of sets as follows:
for each g in Hom(A, X).
For each object B in C we define a contravariant functor
Hom(–,B) : C → Set
as follows:
The functor Hom(–,B) is also called the functor of points of the object B.
Note that fixing the first argument of Hom naturally gives rise to a covariant functor and fixing the second argument naturally gives a contravariant functor. This is an artifact of the way in which one must compose the morphisms.
The pair of functors Hom(A,–) and Hom(–,B) are obviously related in a natural manner. For any pair of morphisms f : B → B′ and h : A′ → A the following diagram commutes:
Both paths send g : A → B to f ∘ g ∘ h.
The commutativity of the above diagram implies that Hom(–,–) is a bifunctor from C × C to Set which is contravariant in the first argument and covariant in the second. Equivalently, we may say that Hom(–,–) is a covariant bifunctor
Hom(–,–) : Cop × C → Set
where Cop is the opposite category to C.
Referring to the above commutative diagram, one observes that every morphism
h : A′ → A
gives rise to a natural transformation
Hom(h,–) : Hom(A,–) → Hom(A′,–)
and every morphism
f : B → B′
gives rise to a natural transformation
Hom(–,f) : Hom(–,B) → Hom(–,B′)
Yoneda's lemma asserts that every natural transformation between Hom functors is of this form. In other words, the Hom functors give rise to a full and faithful embedding of the category C into the functor category SetC (covariant or contravariant depending on which Hom functor is used).
If A is an abelian category and A is an object of A, then HomA(A,–) is a covariant left-exact functor from A to the category Ab of abelian groups. It is exact if and only if A is projective.
Let R be a ring and M a left R-module. The functor HomZ(M,–): Ab → Mod-R is right adjoint to the tensor product functor – R M: Mod-R → Ab.
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