The product of a family {X_i}_(i element I) of objects of a category is an object P = product_(i element I) X_i, together with a family of morphisms {p_i :P->X_i}_(i element I) such that for every object Q and every family of morphisms {q_i :Q->X_i} there is a unique morphism q:Q->P such that p_i °q = q_i for all i element I. The product is unique up to isomorphisms. In the category of sets, the product is the Cartesian product, and in the category of groups it is the group direct product. In both cases, P = product_(i element I) X_i, and p_i :P->X_i is the projection onto the ith factor.
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