I have to be sincere here and say that I still don't get the real difference between a product and a sum.
if I have the trivial order on a set of objects in a category. then the direct limit is a sum and the projective limit is a product.. I really don't see the difference between those two notions: they seem equivalent to me: in a vector space the direct sum of 1 dimensional spaces spanned by the basis vectors could very well be presented by a product.
I hope I was clear in describing my confusion.. I would like to know if even in the trivial case the two notions of direct and inverse limit are not same.