Every locally presentable category is well-powered: since it is a full reflective subcategory of a presheaf topos, its subobject lattices are subsets of those of the latter.

Every accessible category with pushouts (hence also every locally presentable category) is well-*copowered*: this is shown in Theorem 2.49 of *Locally presentable and accessible categories* by Adámek and Rosický, and in Proposition 6.1.3 of *Accessible categories* by Makkai and Paré. The question of whether *all* accessible categories are well-copowered seems to depend on set theory (it follows from Vopenka's principle by Corollary 6.8 of Adámek and Rosický, and implies the existence of arbitrarily large measurable cardinals by Example A.19).

Are all accessible categories well-powered? I have been unable to find a mention of this one way or the other in either of these standard references.