TITLE: Is there a universal way to force the Axiom of Choice to be true?
QUESTION [15 upvotes]: Given a model of set theory $V$ there are various ways to construct a model in which the Axiom of Choice holds, such as Gödel's constructible universe $L^V$ or by using forcing*. I'm wondering if any of these constructions have a nice universal property in the sense of category theory.
In particular are there any adjoints to the inclusion 2-functor $\mathcal{ZFC}\to\mathcal{ZF}$?
(Where $\mathcal{ZF}$ is the 2-category whose objects are either

Models of ZF
Toposes

and whose morphisms are either

Geometric morphisms
Logical functors
Elementary embeddings** 

and where $\mathcal{ZFC}$ is the full subcategory on the objects that obey Choice.)
I think that the concept of the "minimal model" might allow one to construct a right adjoint which is similar in character to the constructable universe. Forcing goes "the other way" by adding sets rather than restricting them, so I suspect it might give a left adjoint.
*Or not? See comments.
**Asaf Karagila notes that there can't exist an elementary embedding between a model where Choice holds and one where it doesn't. So there can't be an adjunction in this case, because its unit or counit would sometimes have to be such a morphism. But perhaps there's some other kind of morphism between models of ZF that does allow an adjunction?

REPLY [4 votes]: Following up on my remarks in the comments, allow me to answer from
the perspective of model-theoretic interpretations of theories. I
view interpretations of theories as providing particularly strong
forms of the desired functors between these categories.
Specifically, an interpretation of one theory $S$ in another
theory $T$, is a uniform way of defining a model of $S$ inside any
model of $T$. Given a model of the latter theory $M\models T$, one
defines a domain $N$ and functions and relations on this domain,
such that with this structure it becomes a model of the first
theory $N\models S$.
Two theories are mutually interpretable, if each of them is
interpreted in the other. So in any model of the one theory
$M\models T$ you can define model of the other theory $N\models S$,
and inside $N$ you can define a model $\bar M\models T$ of the
first theory again.
The thing to notice is that with mutual interpretability, there is
no insistence that these interpretations are inverse of each other,
and it could be that $M$ and $\bar M$ are not much related. Perhaps
this makes them rather un-adjoint-like.
A much stronger notion, therefore, imposes the uniform inverse
requirement. Specifically, theories $S$ and $T$ are
bi-interpretable if they are mutually interpretable in such a way
that the interpreted model $\bar M$ arising in $N$ is isomorphic to
$M$ and furthermore, isomorphic by an isomorphism that is definable
in $M$, and vice versa in the other direction.
The relevance of this for your question is that ZF and ZFC are not
bi-interpretable.
Theorem. Distinct extensions of ZF are never bi-interpretable.
Thus, one cannot transform ZF models and ZFC models into one
another in such a way that they form a bi-interpretation, and I
take this to be a kind of negative answer to a strong version of
the question.
I recently made a blog post providing a proof and further
discussion of this theorem and related matters:


Different set
    theories are never
    bi-interpretable.


(The theorem follows from results of Albert Visser in his 2006
paper, "Categories of theories and interpretations." In addition,
there is a nice automorphism group model-theoretic argument of Ali
Enayat showing for the specific case of ZF and ZFC that they are
not bi-interpretable. Follow the link at my blog.)