Class of compact cardinals implies every accessible category is co-wellpowered

Question

As the title says, I'm looking for the reference that the existence of a class of compact cardinals implies every accessible category is co-wellpowered. It has been stated in Adamek-Rosicky (Locally presentable and accessible categories) that this is proved in Makkai-Paré (Accessible categories: the foundations of categorical model theory), but I couldn't find the proof anywhere. Sorry if this sounds lazy but I don't have a physical copy of the latter book, only a scan which doesn't have OCR so I can't search text.

I've found a proof that powerful images of accessible functors are accessible under the stated assumption, so does co-wellpoweredness follow from this by some argument which I can't see? I'd love a hint in that case.

Any help is appreciated.

Answer 1

In Accessible categories: the foundations of categorical model theory (Makkai, Paré), we find the following:

(part of) Theorem 6.3.8. If there are arbitrarily large compact cardinals, then colimits are uniformly detectable in every accessible category.

Proposition 6.2.2. If pushouts are detectable in an accessible category, then it is co-well-powered.

Detectable and uniformly detectable colimits are defined in 6.2. If colimits are uniformly detectable, then colimits are detectable. What you are looking for follows.