The Mostowski collapse lemma (see here 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)