Discussion
The product does not necessarily exist. For example, an empty product (i.e. is the empty set) 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 it is possible to choose the products in a compatible fashion so that the product turns into a functor . How this functor maps objects is obvious. Mapping of morphisms is subtle, because product of morphisms defined above does not fit. First, consider 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 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 (although some authors use this phrase to mean "a category with all finite limits").
The product is associative. Suppose is a cartesian category, product functors have been chosen as above, and denotes the 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.
Read more about this topic: Product (category Theory)
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