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.