"Vicious Circles" by Barwise and Moss

Hello, I just started reading "Vicious Circles", which is discussion of non-wellfounded or hypersets. It seems like this book could be rewritten from a category theoretic viewpoint at the very least and perhaps even a topos viewpoint. I say topos because I have an intuition that a category with non-wellfounded sets as objects and total functions as morphims is a topos. I am just scratching the surface of this book so it is just an intuition. Or perhaps morphisms should be some other kind of function that preserves structure. Any guidance? Regards, Bill Halchin