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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?

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    $\begingroup$ Every essentially small category is well-powered because well-poweredness is preserved by equivalence and every small category is well-powered $\endgroup$ – John Dougherty May 30 '18 at 4:29
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    $\begingroup$ Well powered is very important in a lot of construction where you want to take limit/colimit indexed by subobjects. For example: one way to define the image of an arrow $f:X \rightarrow Y$ is as the intersection of all subobject of $Y$ in which $f$ factors. This kind of construction only make sense if the category is well powered (other wise you are taking a limit indexed by a class) $\endgroup$ – Simon Henry May 30 '18 at 14:28

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