Bifunctors

In category theory, a bifunctor is a functor that takes two inputs from two categories and is functorial in each input. Given categories C, D and E, a bifunctor F assigns to each pair (X in C, Y in D) an object F(X,Y) in E, and to each pair of morphisms (f: X → X' in C, g: Y → Y' in D) a morphism F(f,g): F(X,Y) → F(X',Y') in E. This assignment respects identities and composition in both variables: F(id_X, id_Y) is the identity on F(X,Y), and F(g2 ∘ g1, f2 ∘ f1) = F(g2, f2) ∘ F(g1, f1).

For fixed Y, the map X ↦ F(X,Y) together with F(f, id_Y) gives a covariant functor C → E, and similarly for fixed X, the map Y ↦ F(X,Y) gives a covariant functor D → E. Thus a bifunctor is functorial in each variable separately.

Variants and variance: some authors define a bifunctor as covariant in both arguments, F: C × D → E. Others allow contravariance in one variable, as in F: C^op × D → E (covariant in the second variable and contravariant in the first). When both arguments are contravariant, one can have F: C^op × D^op → E. The term “bifunctor” is often used for the covariant-in-both-variables case, but the broader language accommodates variance in each argument.

Examples: the Hom functor Hom_C(−,−): C^op × C → Set is a canonical bifunctor, contravariant in the first argument and covariant in the second. The cartesian product functor ×: Set × Set → Set is a bifunctor, as is the tensor product ⊗: Ab × Ab → Ab in the category of abelian groups.

Applications include defining tensor products, enriched category theory, and various constructions that are naturally functorial in two variables. Natural transformations between bifunctors are defined similarly, with functoriality in each variable considered.