Let be a category with some objects and. A product of and is an object together with a pair of morphisms, that satisfy the following universal property:
for every object and every pair of morphisms, there exists a unique morphism such that the following diagram commutes:
The unique morphism is called the product of morphisms and and is denoted. The morphisms and are called the canonical projections or projection morphisms. Above we defined the binary product. Instead of two objects we can take an arbitrary family of objects indexed by some set. Then we obtain the definition of a product. An object is the product of a family of objects if there exist morphisms such that for every object and every -indexed family of morphisms, there exists a unique morphism such that the following diagrams commute for all in : The product is denoted. If = then it is denoted and the product of morphisms is denoted.
Equational definition
Alternatively, the product may be defined through equations. So, for example, for the binary product:
Existence of is guaranteed by existence of the operation.
Commutativity of the diagrams above is guaranteed by the equality =.
The product is a special case of a limit. This may be seen by using a discrete category as the diagram required for the definition of the limit. The discrete objects will serve as the index of the components and projections. If we regard this diagram as a functor, it is a functor from the index set considered as a discrete category. The definition of the product then coincides with the definition of the limit, being a cone and projections being the limit.
Universal property
Just as the limit is a special case of the universal construction, so is the product. Starting with the definition given for the universal property of limits, take as the discrete category with two objects, so that is simply the product category. The diagonal functor assigns to each object the ordered pair and to each morphism the pair. The product in is given by a universal morphism from the functor to the object in. This universal morphism consists of an object of and a morphism which contains projections.
In the category of semi-abelian monoids, the product is given by the history monoid.
A partially ordered set can be treated as a category, using the order relation as the morphisms. In this case the products and coproducts correspond to greatest lower bounds and least upper bounds.
Discussion
The product does not necessarily exist. For example, an empty product is the same as a terminal object, and some categories, such as the category of infinite groups, do not have a terminal object: given any infinite group there are infinitely many morphisms, so cannot be terminal. If is a set such that all products for families indexed with exist, then one can treat each product as a functor. How this functor maps objects is obvious. Mapping of morphisms is subtle, because the product of morphisms defined above does not fit. First, consider the binary product functor, which is a bifunctor. For we should find a morphism. We choose. This operation on morphisms is called cartesian product of morphisms. Second, consider the general product functor. For families we should find a morphism. We choose the product of morphisms. A category where every finite set of objects has a product is sometimes called a cartesian category . The product is associative. Suppose is a cartesian category, product functors have been chosen as above, and denotes a terminal object of. We then have natural isomorphisms These properties are formally similar to those of a commutative monoid; a category with its finite products constitutes a symmetric monoidal category.
Distributivity
For any objects,, and of a category with finite products and coproducts, there is a canonical morphism, where the plus sign here denotes the coproduct. To see this, note that the universal property of the coproduct guarantees the existence of unique arrows filling out the following diagram : The universal property of the product then guarantees a unique morphism induced by the dashed arrows in the above diagram. A distributive category is one in which this morphism is actually an isomorphism. Thus in a distributive category, one has the canonical isomorphism
