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]}
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Intro to Category Theory III: More Mathematical Examples

INTRODUCTION to SET THEORY  DISCRETE MATHEMATICS

INJECTIVE, SURJECTIVE, and BIJECTIVE FUNCTIONS  DISCRETE MATHEMATICS
Transcription
Definition
The product category C × D has:
 as objects:
 pairs of objects (A, B), where A is an object of C and B of D;
 as arrows from (A_{1}, B_{1}) to (A_{2}, B_{2}):
 pairs of arrows (f, g), where f : A_{1} → A_{2} is an arrow of C and g : B_{1} → B_{2} is an arrow of D;
 as composition, componentwise composition from the contributing categories:
 (f_{2}, g_{2}) o (f_{1}, g_{1}) = (f_{2} o f_{1}, g_{2} o g_{1});
 as identities, pairs of identities from the contributing categories:
 1_{(A, B)} = (1_{A}, 1_{B}).
Relation to other categorical concepts
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:
 Hom : C^{op} × C → Set.
Generalization to several arguments
Just as the binary Cartesian product is readily generalized to an nary 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.
References
 ^ Mac Lane 1978, p. 37.
 Definition 1.6.5 in Borceux, Francis (1994). Handbook of categorical algebra. Encyclopedia of mathematics and its applications 5051, 53 [i.e. 52]. Vol. 1. Cambridge University Press. p. 22. ISBN 0521441781.
 Product category at the nLab
 Mac Lane, Saunders (1978). Categories for the Working Mathematician (Second ed.). New York, NY: Springer New York. pp. 36–40. ISBN 1441931236. OCLC 851741862.