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][1], or [nLab][2]) 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][3] 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". 


  [1]: http://www.maths.ed.ac.uk/~aar/papers/maclanecat.pdf
  [2]: http://ncatlab.org/nlab/show/well-powered%20category
  [3]: http://books.google.ru/books/about/General_Topology.html?id=-goleb9Ov3oC&redir_esc=y