I was told that if we have an equivalence of categories $F : \mathcal{A} \rightarrow \mathcal{B}$ with $\mathcal{A}$ abelian, then it is not necessarily true that $\mathcal{B}$ is also abelian.


I would like to know if there are nice examples of an abelian category $\mathcal{A}$ which is equivalent to a non-abelian category $\mathcal{B}$.


Furthermore, are there any conditions over $F$ or $\mathcal{B}$ so that we have "$F : \mathcal{A} \rightarrow \mathcal{B}$ is an equivalence and $\mathcal{A}$ is abelian implies $\mathcal{B}$ is abelian"?



Abelian category equivalent to a non-abelian category
What you were told is wrong, for we have the following:


**Proposition.** *If two categories are equivalent and one of them is abelian, then so is the other.*


A proof (and some related results) can be found in Satz 16.2.4 in H. Schubert, *Kategorien II,* Springer, 1970 (likewise in the English version [https://www.amazon.com/Categories-Horst-Schubert/dp/3642653669](https://rads.stackoverflow.com/amzn/click/com/3642653669), under the same numbering). 


