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.

The product category C × D 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 ${\displaystyle C_{i}}$ indexed by a set ${\displaystyle I}$, ${\displaystyle P=\prod C_{i},p_{j}:P\to C_{j},j\in I}$ 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 ${\displaystyle {\mathsf {Cat}}}$. Thus, for example, $${\displaystyle \textstyle {\mathsf {Fct}}(A,\prod _{i}B_{i})\simeq \prod _{i}{\mathsf {Fct}}(A,B_{i})}$$

where ${\displaystyle {\mathsf {Fct}}}$ denotes a functor category.

Given two functors ${\displaystyle f:C\to D,g:C'\to D'}$, the product ${\displaystyle f\times g:C\times C'\to D\times D'}$ is defined component-wise; that is, $${\displaystyle (f\times g)(x,x')=(f(x),g(x'))}$$ for a pair of objects or morphisms ${\displaystyle x,x'}$. (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