Perhaps I should make more effort to share with the community what I know about stable functors. I am grateful to Walter Tholen for pointing out some old work of his about which I knew nothing. I started writing a much longer survey to send to "categories" but decided it was getting too technical, so I shall just point out one thing. Diers had quite a lot of results about adjoining formal products or coproducts, to get a multiple-valued adjoint. Mike Barr seems to have those results in mind. In my view they make matters far more complicated than they really are. The point in quite simply this: (1) A functor is stable (in my sense) iff it has a left adjoint on each slice. (2) The candidates are the universal maps in the slices. There may be many of them, but they're all as good as each other at being universal maps, coequalisers or whatever. (3) Each object (slice, cocone) only gets to see one candidate. As far as that slice is concerned, that candidate is the real thing. If you can draw a diagram entirely within one slice (for example if it is the square expressing the orthogonality property of a factorisation system) then a coequaliser (or whatever) candidate behaves in exactly the same way as a universal coequaliser would. (4) So long as whatever structure you're interested in (colimits, say) is preserved by pullbacks (as colimits are in a locally cartesian closed category) then yes, the slices are guaranteed to fit together nicely. Hence my word "laminated". I could use "locally", and I'd like a "cartesian" category to be one with pullbacks, but these words are too sullied with overuse. (5) The interesting cases arise when the categories do not have terminal objects. If Max Kelly had dropped this assumption when he investigated "shape transformations" he would have discovered a lot of beautiful results. What's the difference between Diers' "multi" and Lamarche's "poly"? Answer: "poly" is what's true in each slice; this becomes "multi" if the functor preserves equalisers (and hence all connected limits) instead of just wide pullbacks. Of course this is automatic in preorders. Why don't I know of any examples of genuinely multi or poly coequalisers? Because most (all?) of Diers' examples from generalised algebraic theories are full subcategories of the monomorphisms in an equational underlying theory. The inclusion of the formal monos in any factorisation system is a stable functor, whose candidates are the formal epis. If you want to find an example with nontrivial behaviour vis-a-vis coequalisers, you'll have to start with a factorisation system which is completely different from epi-mono. I suggest the final/discrete-fibration factorisation. Alternatively, you can paste slices together. A discussion of multi/poly adjoints, stable functors and candidates will be found in my unpublished paper "The trace factorisation of stable functors". in theory.doc.ic.ac.uk /theory/papers/Taylor/trace.dvi The third section (of three) of this paper discusses adjunctions in the 2-category of stable functors and cartesian transformations. Amongst other things it shows that all such adjunctions are comonadic, and that an ordinary left adjoint belongs to such an adjunction iff it is a discrete fibration. The paper didn't get published because the referee considered this irrelevant technicality, even though it also served as the proof of something the alleged absence of the proof of which he described as "scandalous". It will will never now be published because, as is the nature of research, I can now prove many of the results more easily. Paul Taylor PS I have reverted to magnified computer modern fonts as the default for our latex, even though design size fonts look better. =====================================================================