The Mostowski collapse lemma (see [here](http://en.wikipedia.org/wiki/Mostowski_collapse_lemma) for a quick ref) is one of the key basic tools in the set-theory arsenal. I wonder if the collapse is natural, in the functorial sense. 



> 
> More precisely, is this a
>  reflection from the large category of
>  well-founded models of ZF to the
>  subcategory of transitive models?
> 
> 
> 


My taste would say yes, but I have not thought it through ((apologies if the answer turns out to be trivial).


MOTIVATION: still thinking a bit about the MULTIVERSE Category. If the answer is affirmative, then it makes good sense to simply work with the subcategory of transitive models of ZF, which is certainly more manageable, and simpler to ponder. 


ADDENDUM TO THE MOTIVATION: on a quick after-thought, I partially retract what I just said: there could still be some interest in considering the larger category of not necessarily well-founded models. In this case, perhaps someone could provide some speculations as to this larger cat and what can be found there (exotic models)



Is the Mostowski collapse natural? 
You didn't say what the morphisms in your categories are supposed to be; Trevor assumed you meant elementary embeddings, but you could also have meant mere embeddings, or something else. Nevertheless, unless you make a very strange choice of morphisms, the answer to your question is yes. Not only is the Mostowski collapse a reflection, it's an equivalence of categories. The transitive models constitute a skeleton of the category of all well-founded models; that is, every well-founded model $M$ is isomorphic to exactly one transitive model. Better yet, the isomorphism is uniquely determined by $M$. (All this information is part of the full statement of Mostowski's collapsing theorem.) So, from a category-theoretic point of view, it makes no difference whether you work with arbitrary well-founded models or with only the transitive models. Note, though, that in some situations, non-transitive well-founded models arise naturally, for example as elementary substructures of transitive ones, and in such cases your desire to work only with transitive models would require you to immediately apply the Mostowski collapse as soon as such a model enters your considerations.


