It's a basic principle of category theory that any "good" concept should be preserved by equivalence of categories. However, since Abelian categories are by definition $\mathbf{Ab}$-enriched, we require a bit more structure for our functors – we almost always like them to be additive, and it should preserve at least some form of exactness.
But are there any concepts in Abelian categories that are not preserved by exact equivalences? Or is there some way of proving that any good concept will be preserved by exact equivalences?
On another hand, is it enough to require the functor to be simply an equivalence, in order to guarantee that it is additive and exact?
Best Answer
Equivalences between abelian categories are automatically both additive and exact. This is a good exercise; for a steadily larger series of hints read meditation on semiadditive categories.