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](http://www.maths.ed.ac.uk/~aar/papers/maclanecat.pdf), or [nLab](http://ncatlab.org/nlab/show/well-powered%20category)) 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 ${\sf Mono}(X)$*.



"Uniformly (co)well-powered" categories?
At Sergei Akbarov's suggestion, I'm copying my comment into this answer.


In the presence of the other usual axioms of set theory, including especially the axiom of regularity, the axiom of global choice gives you a well-ordering of the class of all sets. Given such a well-ordering, you can define $\mathcal S\_X$ to contain just the first (with respect to the well-ordering) representative of each isomorphism class of monomorphisms into $X$.


