Wikipedia and Borceux (Handbook of Categorical Algebra, Part I) give the following definitions of subobjects and well-powered categories:

A

subobjectof an object $X$ of a category $\mathsf{C}$ is an equivalence class of the equivalence relation $\equiv$ on the class of all monomorphisms with codomain $X$ where $f \equiv g$ whenever there is an isomorphism $h$ such that $g = f\circ h$.A category $\mathsf{C}$ is

well-poweredif, for any $X \in \mathsf{C}$, all subobjects of $X$ form a set.

There are two different approaches to handling size issues with universes, although they agree on what a (Grothendieck) universe is. One, following Grothendieck, declares that, for a universe $\mathcal{U}$,

a set $X$ is $\mathcal{U}$-small if it is isomorphic to an element of $\mathcal{U}$, a category $\mathsf{C}$ is $\mathcal{U}$-category if its $\mathsf{Hom}$-sets are $\mathcal{U}$-small and it is $\mathcal{U}$-small if it is a $\mathcal{U}$-category and with $\mathsf{Ob(C)}$ being $\mathcal{U}$-small.

This approach depends heavily on Bourbaki set theory and the global choice operator $\tau$. For example, a covariant $\mathsf{Hom}$-functor $\mathsf{Hom}(X,-)\colon\mathsf{C}\to\mathcal{U}\text{-}\mathsf{Set}$ doesn't actually send $Y \in \mathsf{C}$ to $\mathsf{Hom_C}(X,Y)$ but merely to a set in $\mathcal{U}$ isomorphic to it. Another approach (which I'm not sure due to whom, but the book *Higher Categories and Homotopical Algebra* by Cisinski and the paper *Homotopy Limit Functors on Model Categories
and Homotopical Categories* by Dwyer-Hirschhorn-Kan-Smith use this approach) is to declare that

a set $X$ is $\mathcal{U}$-small (or a $\mathcal{U}$-set, the form which I will use here to avoid confusion) if it belongs to $\mathcal{U}$, a category $\mathsf{C}$ is a $\mathcal{U}$-category if objects constitute a subset of $\mathcal{U}$ and all $\mathsf{Hom}$-sets are actually $\mathcal{U}$-sets and it is $\mathcal{U}$-small if it is a $\mathcal{U}$-category whose set of objects is a $\mathcal{U}$-sets.

My question regards the latter approach.

Now an obvious choice for a category to be called $\mathsf{C}$ a $\mathcal{U}$-well-powered is to have a $\mathcal{U}$-set of subobjects as in the definition above. However, this turns out to be useless, since it is not true for most of the categories that we want to be well-powered (again, in the category $\mathcal{U}\text{-}\mathsf{Set}$ of $\mathcal{U}$-sets the set $\bigcup_{Y \in \mathcal{U}} X^Y$ is generally not a $\mathcal{U}$-set). Another approach is to make an exception and relax the definition of $\mathcal{U}$-"smallness", requiring the set of equivalence classes to be $\mathcal{U}$-small in the sense of the first definition. This will now exclude any categories which should be well-powered as when we demanded that the set in question actually belong to $\mathcal{U}$. However, this is not in line with the philosophy of second approach and will probably turn out to be useless in applications if we stick with it.

Now I think of redefining a subobject of $X$ to be *any* monomorphism with codomain $X$ (some books do this, like Riehl's Category Theory in Context) and to say that

a category $\mathsf{C}$ is $\mathcal{U}$-well-powered if, for any $X \in \mathsf{C}$, there exists a $\mathcal{U}$-set of monomorphisms with codomain $X$ containing precisely one monomorphism for each equivalence class of the aforementioned equivalence relation $\equiv$.

This is stronger than requiring each equivalence class to be a $\mathcal{U}$-small, but it seems to work for the usual categories such as those of sets, groups, topological spaces, etc.

What I'm not sure about is how useful will it be if we work with $\mathcal{U}$-sets rather than $\mathcal{U}$-small sets (in particular, if our $\mathcal{U}$-categories have $\mathcal{U}$-sets of morphisms between every two objects rather than simply $\mathcal{U}$-small sets). I thought a little about relation of these approach to the Special Adjoint Functor Theorem, and I think it should work fine. The intuition is this: even if we need to find a monomorphism in this $\mathcal{U}$-set $S$ of distinct representative subobjects with a certain property and we find an object with said property which may not be its element, then we can "replace" it with an equivalent element $S$ which should still satisfy the same property due to the $\equiv$ relation. This is vague, but I think this is what happens in the proof of a Special Adjoint Functor Theorem.

However, I don't know much about well-powered categories and their uses in mathematics, so I need an advice from experts whether this approach leads to any trouble. In particular, I'd like to know if "my" definition of well-poweredness works well with theory of presentable and accessible categories (whose definitions also need to be adjusted if we use universes a-la the paper of Low: see here).