How much set theory does the category of sets remember?
This is not an answer, but it's just more convenient to write a big chunk of text here instead of in a comment. I mostly just have a few small comments (that might be flawed in one way or the other). First, $\textbf{Set}_{\mathcal M}\simeq\textbf{Set}_{\mathcal N}$ holds for any two countable models of ZFC $\mathcal M$ and $\mathcal N$. The reason for this is that for every $\alpha$ we can fix bijections $F_\alpha:\aleph_\alpha^{\mathcal M}\to\omega$ and $G_\alpha:\aleph_\alpha^{\mathcal N}\to\omega$, and then define a functor $e:\textbf{Set}_{\mathcal M}\to\textbf{Set}_{\mathcal N}$ as
$e(f:\aleph_\alpha^{\mathcal M}\to\aleph_\beta^{\mathcal M}):=F_\beta^{-1}\circ G_\beta\circ f\circ F_\alpha^{-1}\circ G_\alpha:\aleph_\alpha^{\mathcal N}\to\aleph_\beta^{\mathcal N}$,
which is both essentially surjective and fully faithful, so an equivalence.
But if either model is uncountable, then an equivalence $e$ would at least require that
- $\text{Card}^V(\kappa)=\text{Card}^V(e(\kappa))$ (since $\hom_{\mathcal M}(1,\kappa)\approx\hom_{\mathcal N}(1,e(\kappa))$)
- $\text{Card}^V((\kappa^\lambda)^{\mathcal M})=\text{Card}^V((e(\kappa)^{e(\lambda)})^{\mathcal N})$ (since $\hom_{\mathcal M}(\lambda,\kappa)\approx\hom_{\mathcal N}(e(\lambda),e(\kappa))$)
So, (1) at least requires that $|\mathcal M|=|\mathcal N|$. Also, $\textsf{Set}_V\simeq\textsf{Set}_L$ would at least imply $\textsf{GCH}$ by (2).