Dear Fred, I keep your notations. The concreteness of A is far from enough to justify the definition you give, namely:
Ever since Eckmann-Hilton, and perhaps even before, the notion of an object G in one category X bearing the structure of an object in some concrete other category A (concrete via U: A -> Sets, say) has been clearly and unambiguously expressed as follows:
The hom functor X(-, G): X^op -> Sets is given a factorization thru' U.
Eckman and Hilton considered only the case when A is a category of essentially algebraic (i.e. definable by projective limits) structures over Sets. In more general cases, it just doesn't work. You can't even prove, for X=A, that an object G of X bears the structure of an object of A. Take for A the concrete category of totally ordered sets and order- preserving functions, with U the obvious forgetful functor. The same is true for A=Fields, A=Topological Spaces, A=Finite Sets, etc., to take very simple examples. Bien amicalement, Jean
If both X and A are concrete, it's perfectly plausible for an object of X to bear the structure of an object in A, and vice versa, and a brief peek at the example of 2 as BA w/ KT_2-space structure and as KT_2-space with BA structure will make short work of understanding how an object may be thought of as "inhabiting both categories at once": indeed, it's that contravariant adjoint pair alone, between A and X, that provides the duality in John Isbell's 1972 approach, where at most one of A and X need be concrete.
HTH. Cheers, -- Fred
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