Dear David, Thanks for your suggestions, and sorry to omit the examples. I'm considering a "category" whose objects are type constructors, given as endofunctions on a base category, and whose arrows are _situated monads_ relating the type constructors. There's no _function_ for monad composition, but perhaps we can obtain a _relation_ by describing conditions for a monad to be a composition of two others. Does that sound reasonable (or like anything else you've seen)? To me this gives your question an entirely different flavor. You might want to post your question again with the above target example added. There is a lot of work on combinig triples, e.g. Manes. My impression is that it is not obvious what it means to combine 2 triples. (One way is the stacking below.) There is some mention of combining triples with distribute laws in Barr, Wells, toposes, triples, and theories, '85, Springer Bicategories may be relevant. MacLane, Pare, coherence for bicategories and indexed categories, JPAA (Journ. of Pure and Applied Alg.) '85 There is a paper by Palmquist, in I think Springer LNM 195, on the double category of adjoint pairs. Double categories are categories internal to the category of small categories. 2-categories are a special case of both bicategories and of double categories. Monoidal categories are a special case of bicategories. Borceux, handbook of categorical algebra, '94, Cambridge may be helpful on some of this. Finitary triples are those commuting with filtered colimits. One can stack them by starting with set to a discrete power, taking a finitary triple on that, forming the category of algebras, taking a finitary triple on that, forming the category of algebras, ... The net result of a finite such stack can also be obtained with just 2 special layers. The 1st has as algebras the sketches in the sense of my `tensor and linear time' and the 2nd follows from the reflector following from the orthogonality. The full development follows, in case you're interested. Maybe I'll read it sometime. The sad truth is that it's hard to write and even harder to get anyone to read it. (However I have found one reader for my writing. Myself. It's amazing what one forgets.) Regards, Jim