As defined, the definition of natural number object seems odd to me, for it does not ensure the multiplicity of its elements. For example, category with exactly one object which has only 1 map satisfies the definition of natural number object , but oddly enough, has only 1 number i.e. 0, for in order to show multiplicity of its elements, one needs to have a structure with multiple elements in the first place, which is not present in this category (as done in this article here: http://www.dcs.ed.ac.uk/home/dt/CT/categories.pdf on page 2).
My question is this: Am I wrong somewhere in concluding that Natural Number object in this special category exists? If so where? If not, why did people then ignore such an important aspect of numbers i.e. their multiplicity?