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 ${\sf Mono}(X)$*.