In the mathematical field of category theory, the product of two categories C and D, denoted 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.
The product category has:
A product of a family of categories is defined exactly the same way.
Just like for sets, a product of a family of categories is characterized by the following universal property. Given categories indexed by a set , satisfy:
Put in another way, a product of a family of small categories is exactly the categorical product of them in the category of small categories . Thus, for example,
Given two functors , the product is defined component-wise; that is,
for a pair of objects and a pair of morphisms . (This product may also be characterized by the universal property similar to that for categories.) This way, we get the functor
It satisfies the tensor-hom adjunction in the sense
where denotes a functor category.
Let be functors. Suppose there is a natural transformation . Then determines the functor
such that
where is the category with two objects and the non-identity morphism . Intuitively, h is a non-invertible homotopy from to . Indeed, define by, for in ,
Conversely, given , we get by and .
A functor whose domain is a product category is called a bifunctor. A bifunctor can be defined in each variable separately in the following sense:
For example, consider . For each fixed in , we have the functor
by pullback; i.e., goes to the function
defined by . On the other hand, is defined by pushforward; i.e., . Clearly, these two functors commute (the associativity of composition) and so, by the proposition, we get the functor called the Hom functor
which is explicitly given as:
There is a similar result for natural transformations between bifunctors: