Some considerations, not a full answer (yet).

- In *Accessible categories and models for linear logic*, at page 2, Barr claims that every accessible category is well powered. He even claims that is observed in the classical reference by Makkai-Parè. I did not manage to find it.

- A locally small category with finite intersections of subobjects and a (strong) generating set is well-powered, this appears in Johnstone [Sketches of an elephant, Remark A1.4.17]. Thus when an accessible category has finite intersections, it is well powered.

I still believe that every accessible category is well powered, and I am looking through the literature.

- Observe that the statement that appears in Rosicky-Adamek on page 2, namely *every locally small category with a strong generator is well-powered* is wrong, as proved by the following [counterexample][1].


  [1]: https://math.stackexchange.com/a/254794/36248