This simultaneously captures the concept of a product of sets, posets, groups, [-topological_space topological spaces] etc. In addition, like any universal construction, this characterization does not differentiate between isomorphic versions of the product, thus allowing one to abstract away from an arbitrary, [-specific_construction_category_theory specific construction].
Definition
Given a pair of objects $~$X$~$ and $~$Y$~$ in a category $~$\mathbb{C}$~$, the product of $~$X$~$ and $~$Y$~$ is an object $~$P$~$ along with a pair of morphisms $~$f: P \rightarrow X$~$ and $~$g: P \rightarrow Y$~$ satisfying the following universal condition:
Given any other object $~$W$~$ and morphisms $~$u: W \rightarrow X$~$ and $~$v:W \rightarrow Y$~$ there is a unique morphism $~$h: W \rightarrow P$~$ such that $~$fh = u$~$ and $~$gh = v$~$.
Comments
Eric Bruylant
Would Product (mathematics) be an appropriate name, or does category theory's use of the term point to only a subset of the things product can mean?
Mark Chimes
Eric Bruylant Whether Product (mathematics) is appropriate really depends if you're asking a category theorist (who would say yes) or not . ;-)
In seriousness, specific kinds of products include cartesian products, products of algebraic structures, products of topological spaces and the most well known: product of numbers. All of these are special cases of the categorical product (if you pick your category right), but I can imagine someone wanting to look up 'product' as in multiplication and getting hit with category theory.
I don't know. It's a matter of taste I suppose. I get the idea that category theory is not yet quite widely-known enough for this to be considered "the" definition by most mathematicians, but if other contributors feel it should be given that status I certainly won't complain. I just thought this was the safer approach.
See, for example product on Wikipedia.
Mark Chimes
Yeah, I think keeping it as it is now is probably the best way of following the "one idea per page" methodology. The page on Products (mathematics) can have this page as child.