My question concerns categorical "proofs" or "guessing" of arithmetical identities using category theory. I’m not at all a specialist of CT but I try to put my hands on and to understand something in this fascinating subject. Trying to prove something concrete, the exercise was to prove the identity
$$ \gcd(n,m) \cdot \operatorname{lcm}(n,m) = n\cdot m$$

Using the concepts of product and coproduct in a specific category (objects are integers and arrows is "divisibility") I've been able to prove it with very minimalistic assumptions, only the notion of divisibility of integers. No use of prime numbers or unique factorization with primes or other specific theorems like Bezout theorem, etc...

I'm wondering if someone has other examples like that one. Someone gave me the reference

* Marcelo Fiore, Tom Leinster, *Objects of Categories as Complex Numbers*, Advances in Mathematics **190** (2005), 264-277, doi:[10.1016/j.aim.2004.01.002](https://doi.org/10.1016/j.aim.2004.01.002), arXiv:[math/0212377](https://arxiv.org/abs/math/0212377),

which is interesting but it doesn't really answer my question. In that paper, if I've correctly understood, they show that if something is true with complex numbers it is true in more general categories.

I would be interested if someone has some "simple" examples of concrete applications of CT proving or "guessing" arithmetical identities

Thanks a lot for any clue