Hom Functor - Formal Definition

Formal Definition

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 and B in C we define two functors to the category of sets 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. The notation Hom_C(–,–) is sometimes used for Hom(–,–) in order to emphasize the category forming the domain.