When and are elements of sets, the pairing of and is the ordered pair .
It is natural to extend this to generalised elements in any category with binary products.
For products of higher arity, one can say tripling, quadrupling, etc, or just tupling.
Let and be objects of some category , and suppose that the? product exists in .
Let be some object of , and let and be morphisms of . Then, by definition of product, there exists a unique morphism such that the obvious diagrams commute.
If we think of and as -indexed elements of and , then is a -indexed element of .
If is the category of sets and is the point, then and are simply elements, in the usual sense, of and ; then is an element of , the usual ordered pair .
If and are each , with and each the identity morphism on , then the pairing is the diagonal morphism .
Pairings versus products
Since taking products (when these always exist) is a functor, we can apply this operation to any two morphisms. That is, if and are morphisms in a category , and if the products and exist, then we have a morphism . This is not the pairing , for which the source is always .
A pairing is the composite of a product and a diagonal morphism:
conversely, a product is a pairing of two composites:
If and are each terminal, however, then and are the same global element of . Thus, both product morphisms and pairings are generalisations of ordered pairs in Set.