The notions of sums and products one learns first with natural numbers.
In abstract algebra, there are related notions - "Direct sum" and "Direct product".
Now in Category theory, these notions have been generalized in an interesting way that in many categories, we see that the co-product or product of two objects is the most natural one which we can see in the category - for example, look these notions categories of sets, abelian groups, modules, etc.
Now my question is a "natural one":
Is there a natural category with objects being natural numbers, and in which, the co-product (sum) of $m$ and $n$ is $m+n$, and product of two objects $m$ and $n$ is $m.n$?
While comparing direct sums and direct products of modules or vector space, one intuitively thinks that product is bigger than the sum, which also happens in case of natural numbers: $m.n\geq m+n$. This way of look at products and co-products strikes the question in mind which is written above.
Take a natural number $n$ to be the set $\{0,\ldots,n-1\}$. This is actually the most common ZFC definition of the natural numbers. Then, simply take the category $\mathbb N$ to have natural numbers as objects, and functions as morphisms. Then $n \sqcup m = n + m$ and $n \times m = nm$.
This is the same as/reflects the fact that $|X \sqcup Y| = |X| + |Y|$ and $|X \times Y| = |X||Y|$ for all sets $X,Y$.