It is sometimes emphasized that a "concrete category" is not a property of a category $C$, but rather a structure, i.e. a faithful functor from $C$ to $Set$. Thus, When people talk about a concrete category $C$ they *really* mean $C$ together with some implicitly defined and naturally understood functor from $C$ to $Set$ (most commonly, when the objects of $C$ are sets with some extra structure and the morphisms are some functions between those sets). This emphasize suggests that for a given category $C$, there might be several inequivalent concretizaions (of course there might also be none, but this is less important for this discussion) where I take "equivalent" to mean the two concretizing functors are naturally isomorphic (this seems natural, but is it the "correct" definition?).

In light of this, I would like to see an interesting as possible example of a category $C$ with two inequivalent concretizations. I guess there are tailor made examples with possibly even a finite category, though I most confess I didn't try to find such myself, so it might be also interesting to see such an example, but the most satisfying example would be of a category of some sort of "real life" mathematical structure with two meaningful inequivalent concretizations each giving some different intuition about the category (perhaps this is to much to ask, but it is only to clarify what I mean by "interesting").

interestingexamples. In other words, examples where one might blithely refer to "the" underlying-set functor as the "obvious" one, but on reflection it's not so obvious: that's how I read KotelKanim's second sentence. – Todd Trimble♦ Oct 27 '11 at 17:24