Let's say we have category with one object $N$ and infinite number of arrows, which are named as natural numbers, with the same law of composition, where $id$ arrow 0.
I try to understand 1) if categorical product/coproduct could be defined in such category and 2) if it could have properties of usual binary operations on natural numbers?



Is product/coproduct in category with only one object possible?
The category you describe lacks any interesting products and coproducts. But there is a category with only one object that has very interesting products and coproducts. Namely, let $A$ be a [$III\_1$ factor](https://en.wikipedia.org/wiki/Von_Neumann_algebra#Type_III_factors), and consider the category with only one object and endomorphisms of that object indexed by elements of $A$ (and composition given by multiplication). It is not obvious, but this category has both products and coproducts.


