Can a category be enriched over abelian groups in more than one way? An $\mathbf{Ab}$-category is a category enriched over the category of abelian groups. What is an example of a category that can be enriched over abelian groups in more than one way?
An abelian category is a very special type of $\mathbf{Ab}$-category. So can a category be given the structure of an abelian category in more than one way, or is being abelian unique and hence a property of the category?
 A: Just to add to Wojowu's answer, here is an explicit example of a 1-object category $C$ which is enriched over abelian groups in more than one way.
The morphism set of $C$ is $\mathbb Z$. Note that $\mathbb Z$ is a monoid under usual multiplication, and this is how we define composition.
One way to make $C$ into a preadditive category is the obvious one, where $m+n$ is defined as usual.
For an alternate way, note that, by the uniqueness of prime factorization we have an isomorphism between the multiplicative mononid of $\mathbb Z$ and the commutative monoid $\{-,+\} \times P$ where $\{-,+\}$ is a finite simple group of order 2 and $P$ is the free commutative monoid on the set $\mathcal P$ of primes. Any permutation of $\mathcal P$ induces an automorphism of $P$ and hence of our category $C$. We can then conjugate addition by such an automorphism to get a distinct preadditive structure on our category $C$.
A: You can easily find examples among categories with one element: a category with one element is a (multiplicative) monoid, and $Ab$-enrichment over it is a choice of an addition which turns it into a ring. And there can be multiple such additive structures: you can for instance consider pullback along a permutation which preserves multiplication but not addition.
More interestingly, if we instead work with additive categories rather than preadditive ones (meaning we require all finite products to exist), then it turns out $Ab$-enrichment is necessarily unique. This in particular applies to abelian categories. Indeed, given two morphisms $f,g:A\to B$, we can characterize $f+g$ as the composition
$$A\to A\oplus A\to B\oplus B\to B,$$
where the first map is the diagonal map, the last map is the codiagonal, and the middle map is the map which you can think of being $f,g$ on the components. See Wikipedia for some more details.
