Say Fcc is the category of all (small) finitely complete categories with finite-limit-preserving functors as morphisms, and Int is the functor which takes an object C of Fcc to the (finitely complete) category of internal category objects* and internal functors in C, and takes an arrow f : C -> D of Fcc to Int(f) defined by Int(f)(H, d_0, d_1, comp) := (f(h), f(d_0), f(d_1), f(comp)). Does Int have an obvious right adjoint, or does it at least preserve colimits? ---Jason * I honestly don't know what the most official definition of these are, (and I can't get to a library for a couple weeks) but I had the one in mind where an internal category is an object H of C, and arrows d_0,d_1 : H -> H and comp : H \times_H H -> H satisfying the appropriate diagrams, and internal functors are the obvious thing)