Is the Mostowski collapse functorial?

Question

It is a standard result that given an extensional, irreflexive, regular, well-founded relation $R$ on a class $X$ one can magick out its (transitive) Mostowski collapse $\overline{X}$ together with an isomorphism $(\overline{X},\in) \cong (X, R)$. My question is: is this construction in any way functorial? Can we see this as a endofunctor on, say, the category of classes and extensional, irreflexive, regular and well-founded relationships, or some more complicated category?

Answer 1

Shifting attention from classes to sets for simplicity - in particular, so that we're staying in the realm of genuine (and indeed locally small) categories - Mostowski says the following:

If $R$ is a well-founded, extensional, and irreflexive relation on a set $X$, then there is a unique transitive set $T_{R,X}$ such that $(T_{R,X};\in)\cong (X;R)$, and moreover that isomorphism is unique.

So let $\mathcal{S}$ be the category of all sets equipped with such relations, and let $\mathcal{T}$ be the subcategory of $\mathcal{S}$ consisting of all transitive sets equipped with $\in$. Then Mostowski says that $\mathcal{T}$ is a skeleton of $\mathcal{S}$ in a very nice way. In particular, the Mostowski collapse does indeed give a functor $\mathcal{S}\rightarrow\mathcal{T}$.

And looking back at clase vs. sets, size issues don't add any interesting content here - they just make the language a bit more annoying.