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).

This property seems to be weaker than 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 ${\sf Mono}(X)$.

(from the axiom of choice it doesn't follow that such a map automatically exists).

My questions:

1. Which name do people use for this stronger property of "well-poweredness"?
2. Does anybody know examples of well-powered categories which are not well-powered in this stronger sense?
3. The same for the dual property of "co-well-poweredness".