nLab bifunctor

This article is about functors on product categories. For morphisms between bicategories see 2-functor and pseudofunctor.

Definition

By a bifunctor (short for binary functor, that is $2$-ary) or functor of two variables is simply a functor whose domain is a product category:

For $C_1$, $C_2$ and $D$ categories, a functor

$$F \;\colon\; C_1 \times C_2 \longrightarrow D$$

is also called a bifunctor from $C_1$ and $C_2$ to $D$.

Remark

(terminology)
While the term bicategories is used for (weak) 2-categories, the terminology “bifunctor” is not used in this context, instead one speaks of pseudo-functors or 2-functors between bicategories/2-categories.

In fact, even for the sense of a functor of 2 variable, the term “bifunctor” may not be used as frequently anymore as it used to, except maybe in parts of computer science and in model category-theory (cf. Quillen bifunctor).

Examples

Famous bifunctors are

• the hom functor

  $$Hom(-,-) : C^{op} \times C \to Set$$

  on any locally small category $C$, or if $C$ is a closed category, the internal hom functor

  $$[-,-] : C^{op} \times C \to C \,.$$

• on every monoidal category $(C, \otimes)$ the tensor product functor

  $$\otimes : C \times C \to C$$

A bifunctor of the form $D^{op} \times C \to Set$ is called a profunctor.

Related concepts

• extranatural transformation

• binary function, bilinear map, multilinear map

• binary morphism

• bifunctor, two-variable adjunction, Quillen bifunctor

  end/coend

• multifunctor

References

In the generality of enriched category theory (hence for enriched functors on enriched product categories):

• Samuel Eilenberg, G. Max Kelly, §III.4 of: Closed Categories, in: Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer (1966) 421-562 [doi:10.1007/978-3-642-99902-4]

• Richard Garner, and Ignacio López Franco?. “Commutativity.” Journal of Pure and Applied Algebra 220.5 (2016): 1707-1751.

• Nicola Gambino, Richard Garner, and Christina Vasilakopoulou. “A unified treatment of commuting tensor products of categories, operads, symmetric multicategories and their bimodules.” arXiv:2511.14402 (2025).