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").



two essentially different concretizaions
One of course has many examples. To define the terms, a *concretization* of a category $C$ should be a faithful functor $C \to \mathrm{Set}$. In examples, concretizing functors are never full, so I will not ask this, but I do not mind asking, say, that the concretizing functor reflect isomorphisms. Maybe you want moreover for "concretization" to have some adjointness properties with some "free" or "discrete" functor $\mathrm{Set} \to C$, but maybe not. But even if you do ask for this, then there are still many such functors. For example, take any concretizing functor $F : C \to \mathrm{Set}$; then probably the functor $2\times F$, which assigns to $x\in C$ the set $F(x) \sqcup F(x) = \lbrace 0,1\rbrace \times F(x)$, is also concretizing.


An example that comes up in nature is the following. I can concretize the category of finite-dimensional Lie algebras by assigning to each Lie algebra the underlying set of its underlying vector space. Instead, I can embed the category of finite-dimensional Lie algebras into the category of Lie groups (by assigning to each Lie algebra its connected simply-connected group), and then I can concretize the category of Lie groups by assigning to each its underlying set. To see that these functors are not isomorphic, consider their actions on any non-trivial morphism $\mathbb R \to \mathfrak{so}(3)$. Under the first concretization, this morphism turns into an injection from the set $\mathfrak c$ with cardinality continuum into itself. Under the second concretization, the morphism also becomes a map $\mathfrak c \to \mathfrak c$, but it is not an injection.


