Product (category theory) 

Suppose C is a category, I is a set, and for each i in I, an object X_i in C is given. An object X, together with morphisms p_i : X → X_i for each i in I is called a product of the family (X_i) if, whenever Y is an object of C and q_i : Y → X_i are given morphisms, then there exists precisely one morphism r : Y → X such that q_i = p_ir.

The above definition is an example of a universal property; in fact, it is a special limit. Not every family (X_i) needs to have a product, but if it does, then the product is unique in a strong sense: if p_i : X → X_i and p'_i : X ' → X_i are two products of the family (X_i), then there exists a unique isomorphism r : X → X ' such that p'_ir = p_i for each i in I.

An empty product (i.e. I is the empty set) is the same as a terminal object in C.

If I is a set such that all products for families indexed with I exist, then it is possible to choose the products in a compatible fashion so that the product turns into a functor C^I → C. The product of the family (X_i) is then often denoted by ΠX_i, and the maps p_i are known as the natural projections. We have a natural isomorphism[?]

<math>\operatorname{Mor}_C\left(Y,\prod_{i\in I}X_i\right) \simeq \prod_{i\in I}\operatorname{Mor}_C(Y,X_i)</math>

If I is a finite set, say I = {1,...,n}, then the product of objects X₁,...,X_n is often denoted by X₁×...×X_n. Suppose all finite products exist in C, product functors have been chosen as above, and 1 denotes the terminal object of C corresponding to the empty product. We then have natural isomorphisms[?]

<math>X\times (Y \times Z)\simeq (X\times Y)\times Z\simeq X\times Y\times Z</math>

<math>X\times 1 \simeq 1\times X \simeq X</math>

<math>X\times Y \simeq Y\times X</math>