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](https://mathoverflow.net/q/324621/49) 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?



Accessible functors not preserving lots of presentable objects
An example is given in my paper with Tibor Beke,



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> *Abstract elementary classes and accessible categories*, Annals Pure Appl. Logic **163** (2012), 2008-2017, doi:[10.1016/j.apal.2012.06.003](https://doi.org/10.1016/j.apal.2012.06.003), arXiv:[1005.2910](https://arxiv.org/abs/1005.2910).
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see Remark 3.2(4). This is what Reid Barton indicated.


