Every [locally presentable category](https://ncatlab.org/nlab/show/locally+presentable+category) is [well-powered](https://ncatlab.org/nlab/show/well-powered%20category): since it is a full reflective subcategory of a presheaf topos, its subobject lattices are subsets of those of the latter.


Every [accessible category](https://ncatlab.org/nlab/show/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.



Is every accessible category well-powered?
It seems to me that every category with a small set of *dense* generator is well powered. In particular accessible categories are well powered.


Dense generator means that you have a small full subcategory $C \subset A$ such that the induced nerve functor $A \rightarrow \widehat{C}$ is fully faithful. This functor send monomorphisms to monomorphisms so, as it is fuly faithful, isomorphisms class of subobjects in $A$ are a subset of the isomorphisms class of subobjects in $\widehat{C}$ (which is well powered). 


Note: this does not conflict with Ivan's answer, Strong generator only means the nerve is faithful and conservative and this is not sufficient to deduce that the Nerve is injective on iso class of mono. One can have two mono $A \subset X$ and $B \subset X$ with isomorphic image under the nerve functor, if none of them is included in the other then you can't deduce that $A$ is isomorphic to $B$.


Of course if one add the assumtpion that intersection of subobjects exists then one can use their intersection to compare them using only conservativity. i.e. if we have intersection of subobjects, a set of strong generator is enough, as mentioned by Ivan.


