The product category 
of categories 
and 
has object pairs 
and morphism pairs 
. Identities and composition
are defined componentwise. Its projections to 
and 
make it the categorical product
of the two categories in the category of small categories.
A bifunctor on 
and 
is a functor whose domain
is their product category.
Product Category
See also
Bifunctor, Category, ProductExplore with Wolfram|Alpha
References
Mac Lane, S. Categories for the Working Mathematician. New York: Springer-Verlag, 1971. https://doi.org/10.1007/978-1-4612-9839-7.Cite this as:
Weisstein, Eric W. "Product Category." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ProductCategory.html