More precisely, is every concrete category C isomorphic to a category C' of small categories such that the morphisms between two elements of C are precisely the functors between their images in C'?


At some point I started adopting this point of view as a philosophy without ever bothering to actually verify it. 



Can the objects of every concrete category themselves be realized as small categories?
No. There is no full subcategory of Cat equivalent to • ⇉ •, because for any two objects C and D of Cat, either both Hom(C, D) and Hom(D, C) are nonempty, or one of C and D is the empty category and then one of Hom(C, D) and Hom(D, C) is empty and the other is the category •.


