TITLE: Accessible functors not preserving lots of presentable objects
QUESTION [11 upvotes]: Let $F:\cal C\to D$ be an accessible functor between locally presentable categories.  By Theorem 2.19 in Adamek-Rosicky Locally presentable and accessible categories, there exist arbitrarily large regular cardinals $\lambda$ such that $F$ preserves $\lambda$-presentable objects.  It is tempting to expect that $F$ should preserve $\lambda$-presentable objects for all sufficiently large $\lambda$, but that is not what the theorem says.  However, I do not know a counterexample showing that the stronger claim fails.  (For instance, this question asks about this property when $F$ is the pullback functor, and has no answer yet in the general case.)
What is an example of an accessible functor $F$ between locally presentable categories for which there exist arbitrarily large regular cardinals $\mu$ such that $F$ does not preserve $\mu$-presentable objects?

REPLY [12 votes]: An example is given in my paper with Tibor Beke,

Abstract elementary classes and accessible categories, Annals Pure Appl. Logic 163 (2012), 2008-2017, doi:10.1016/j.apal.2012.06.003, arXiv:1005.2910.

see Remark 3.2(4). This is what Reid Barton indicated.