I am reading the introductory lessons on Category Theory on wikiversity, and they discuss free monoids here: <https://en.wikiversity.org/wiki/Introduction_to_Category_Theory/Monoids> 

At the bottom they prove that the monoid of integers is not free, and their definition of a free monoid is that whose every arrow can be uniquely represented as a composition of arrows from a finite collection S of arrows. So, to disprove that the integers are a free monoid, they try to show that the arrow 1 can be generated by two distinct compositions of generators:

"If the set of generators contains the number 1, then it can't contain any other positive integer, since every positive integer is a finite sum of 1's. It can't contain any negative numbers either, since any negative number plus a finite sum of 1's is 1.

On the other hand every set of generators must generate number 1, so 1 is the finite sum $(s\_1 + s\_2 + ... + s\_n)$ of generators. But a finite sum of finite sums is a finite sum, so every positive generator $s\_0$ can be composed as $s\_0 = (s\_1 + ... + s\_n)\_1 + (s\_1 + ... + s\_n)\_2 + ... + (s\_1 + ... + s\_n)\_{s\_0}$

For a negative generator $-s\_0$ we have $-s\_0 + (s\_1 + ... + s\_n)\_1 + (s\_1 + ... + s\_n)\_2 + ... + (s\_1 + ... + s\_n)\_{s\_0+2} = 1 = (s\_1 + ... + s\_n)$

that gives two different compositions for the number 1."

EDIT: As the comments point out my source of knowledge is not ideal, so if there is a proof with better (standard?) notation I'd appreciate it if anyone can share it or can offer any hints to it.