My question concerns the notion of naturality defined in Section 12.3 of Hodge's (longer) Model Theory. See the addenda for the definition.
Hodges proves a result that if an algebraic construction of $B$ from $A$ is not natural, then (if Pincus's transfer principle from ZFA to ZF is applicable) there is a model of ZF with $A$ but with nothing isomorphic to $B$. An example of such a construction is one of the algebraic closure $B$ from a field $A$. One is naturally interested in whether its converse obtains in certain circumstances.
The textbook cites two papers by the author himself and Shelah (Naturality and definability, I and II) which study a similar question but concern global choice principles. The papers study the converse as well, but I do not believe that they answer the question above.
Is there a paper that address the question whether Hodges's notion of naturality implies that the construction can be done without AC, at least in some cases?
Addendum. The following is Hodges's definition of naturality. The setting is that $B$ has extra nonlogical symbols that $A$ doesn't have, and a unary predicate symbol $P$ is among the extra symbols. The subset $P^B$ is assumed to induce a substructure of $B$ that happens to be $A$. Moreover, we assume that every automorphism on $A$ extends to one on $B$, i.e., there is a (set-theoretic) map (called a section) $\mathrm{Aut}(A) \to \mathrm{Aut}(B)$. Under all these assumptions, $B$ is natural over $A$ if the section is a group homomorphism, i.e., a splitting.
Addendum II. I guess that Hodge's notion of naturality translates to that in category theory as follows. Let $C(M)$ stand for the category of structures isomorphic to a structure $M$ whose morphisms are the isomorphisms. The construction in question can be thought of as a functor $F : C(A) \to C(B).$ In this setting, the construction is natural in the sense of Hodges iff the inclusion maps $\eta_A : A \to F(A)$ forms a natural transformation from the identity functor on $C(A)$ to $F$.