Let $C$ be a category and for each object $X$ let us denote by ${\sf Mono}(X)$ the category of all monomorphisms in $C$ going to $X$ (i.e. monomorphisms which have $X$ as range -- I hope it is clear how morphisms in ${\sf Mono}(X)$ are defined).

$C$ is said to be *well-powered* (see MacLane, or nLab) if for each object $X$ the category ${\sf Mono}(X)$ has a "small" skeleton $S$ (i.e. a skeleton, which is a *set*, not a proper class).

Is this property equivalent to the following one:

*there is a map $X\to S_X$ which assigns to each object $X$ a "small" skeleton $S_X$ of the category ${\sf Mono}(X)$*.

setof itself, namely the set of those members of the subclass that have the smallest rank (in the sense of the cumulative hierarchy of sets) among the members of that subclass. In your situation, you'd have the class of monomorphisms partitioned into isomorphism classes. – Andreas Blass Jul 21 '13 at 4:33