Category Theory and Hereditarily-Finite Sets

Hello Cat Theory Community, Hereditarily-finite sets are becoming increasingly more popular in computer science research. 1) What kind of interesting categories exist where an "object" is a hereditarily-finite set plus some structure on the set and a "morphism" would be a structure-preserving function. (I can think of the obvious subcategory of SET and also the category where an object is a hereditarily-finite set together with a "SET" endomorphism on that set, but neither of these categories would have interesting or useful properties, in my opinion) 2) What kind of papers can I read on this subject? Regards, Bill Halchin