Sorry, I used K/S for an abbreviation of what was called Kuratowski until someone pointed out that it was due to Sierpinski :an object whose mark belongs to the smallest sub-semilattice of its power set which contains the singleton map, or in case there is an NNO an object which in a suitable sense is locally enumerable by the segment under a section of the NNO . While the K/S definition is right for the construction of the object classifier over an arbitrary base topos (as Gavin showed) and hence for classifiers for various kinds of finitary algebras over an arbitrary base topos, still the theory of it in the last 25 years of topos theory seems to mainly be justified by formal analogy and/or independence relative to abstract set theory (=topos with choice). However there are important uses of "finiteness" in algebraic geometry and differential topology (where topos theory after all started) : Consider a ringed topos E,R . For example, the sheaves on an algebraic variety or on a Cinfty manifold. Within the abelian category of R-modules in E, we need to single out two important subcategories FAC (Serre 1955)=coherent sheaves..these tend to be an abelian subcategory and tend to vary covariantly as one E,R is mapped to another E',R' (thus give rise to an extensive K-homology) and vector bundles , which one thinks of as a finite-dimensional vector space varying smoothly over the base space of E ,so they cry out for internalization ; in algebraic geometry these are identified with locally FINITELY free R-modules... they vary contravariantly with E,R (so give rise to K-cohomolgy rigs which act on the FACs,ie intensives acting as densities on the extensives; with further conditions on E,R one can at the level of the riNgs generated by these rigs define a sort of Radon/Nikodym derivative via an alterating sum of Tors , but in general the covariant abelian category FAC and the contravariant tensored category Vect are distinct...The "derived category" of E,R (now allegedly replacing homological algebra in complex analysis and C*-algebra theory) should be the derived category of one of these two linear categories (here I mean dc in the linear sense..nonlinear "derived categories" are more like the stable homotopy of E)) Already the intuitionists speculated about (in effect) subobjects of K/S objects, and it seems we need something of the sort perhaps a category of finites closed under subquotient in order to define the notion of eg finitely-generated R-module in a way which not merely mimics abstract set theory but actually captures the vector bundles . Perhaps it will be easier if E itself satisfies a noetherian condition. It would be best if the desired content could be entirely int- ernalized to E,R but perhaps it is really relative to a base S,K..but perhaps without restriction on S ?? I hope this clarifies the problem. Sincerely Bill