Is a pre-additive structure on a category $\mathcal{C}$ necesarrily unique? If $\mathcal{C}$ is an additive category, i.e. it is $\mathbf{Ab}$-enriched and moreover it admits finite biproducts, it is quite well-known that the additive structure is uniquely determined by internal properties of $\mathcal{C}$. All the proofs I know make use of the existence of biproducts so I suppose this fact is crucial. However I wondered:

Are there easy accesible examples of categories $\mathcal{C}$ which can be enriched over the category of abelian groups in several non-equivalent ways?


It suffices to give an example of two rings (with unit but not necessarily commutative) that have isomorphic multiplicative monoids but non-isomorphic additive groups: use the identification of rings with one-object preadditive categories.
So consider $\mathbb{Z}$. Its multiplicative monoid is (by the fundamental theorem of arithmetic) isomorphic to $\{ 0 \} \cup \{ \pm 1 \} \times \mathbb{N}^\mathbb{N}$. But the same is true of the polynomial ring $\mathbb{F}_3 [x]$ (because polynomials over any field form a UFD), and we are done.
