# Product category

In the mathematical field of category theory, the **product** of two categories *C* and *D*, denoted *C* × *D* and called a **product category**, is an extension of the concept of the Cartesian product of two sets. Product categories are used to define bifunctors and multifunctors.^{[1]}

For small categories, this is the same as the action on objects of the categorical product in the category **Cat**. A functor whose domain is a product category is known as a bifunctor. An important example is the Hom functor, which has the product of the opposite of some category with the original category as domain:

Just as the binary Cartesian product is readily generalized to an *n*-ary Cartesian product, binary product of two categories can be generalized, completely analogously, to a product of *n* categories. The product operation on categories is commutative and associative, up to isomorphism, and so this generalization brings nothing new from a theoretical point of view.