product functor, abstractly

Hi, In any category with products, one can define the diagonal morphism (from an object X to the object X*X). In the special case of the category of small categories, the above definition gives rise to the diagonal functor (from a category C to the category C*C). How can I in a similar manner define the product functor (from the category C*C to the category C)? I.e. in any category with products, is there a morphism from X*X to X that would give rise to the product functor in the special case of the category of small categories? I guess that I need to assume something on X, but I cannot find the right assumption. Thanks for any help! [For admin and other information see: http://www.mta.ca/~cat-dist/ ]