A category $\mathcal{C}$ is called *well-powered* if for any $X \in \mathcal{C}$ the class $\mathrm{Sub}(X)$ of subobjects of $X$ is a set. It is called *essentially small*, if the class of isomorphism classes of objects is a set.

**Question.** Are there essentially small categories which are not well-powered? (My problem here is that being isomorphic as subobjects is stronger than just being isomorphic, so a priori there may be more classes of subobjects).

**Extra question.** When defining the notion of noetherian/artinian for objects of a category, no one seems to care if the class of subobjects is a set. Is there an issue or can one work just fine with partially ordered classes? (E.g. the theorem that having a composition series is equivalent to noetherian and artinian, does it hold?). If so, then why is well-powered important? When do I have to be careful?