In his comments to both cody and Nik Weaver regarding his answer to user7280899's mathoverflow question "What kind of foundation are mathematicians using when proving metatheorems?", Mike Shulman writes:
I care more about not relative consistency, but "interpretability": that you can directly construct a model of one theory from the other theory in a relatively straightforward way. This tells us how to actually relate (at least in principle) mathematics formalized in the two theories. If someone proves a theorem in $ZFC$, then I understand directly what it means in $SEAR$ [Sets, Elements, And Relations] as a theorem about hereditarily well-founded relations, and vice versa: a theorem in $SEAR$ is a theorem in $ZF$ about the category of sets...Put differently, consider the category of models of $ZF$ and the category of models of $SEAR$. To say $Con(ZF)$ $\Leftrightarrow$ $Con(SEAR)$ is to say that if one of these categories has an object, so does the other. I would much rather know that these categories are related by an adjunction or an equivalence, even if I need a stronger metatheory.
Which brings me to my question:
How are material set theory and structural set theory related from the point of view of category theory? More specifically, how is the category of models of $ZF$ related to the category of models of $SEAR$ from the point of view of category theory? And what is the significance of this relation to the foundations of mathematics (since Set Theory and Category Theory have been considered (per se) to be alternate foundations for mathematics)?