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_{i}r.
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'_{i}r = 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[?]
If I is a finite set, say I = {1,...,n}, then the product of objects X_{1},...,X_{n} is often denoted by X_{1}×...×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[?]
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