Product

Let be a Category and
A product of and is an Object
together with Morphisms and
with the universal property that
for any with maps and
there exists a unique
such that we have a Commutative Diagram

When is Locally Small, we can define a Functor
defined by

If this is Representable, let be its Representation.
i.e. is a Natural Isomorphism
Note that is the Universal Element,
consisting of where and
and thus is the product of and .