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, under the same numbering).
