Dear Mike, Why the fuss about allowing operations that don't preserve the order. Surely no such operations could arise from algebras for triples over POS. When handed a category B that has a cartesian closed structure, and (in your notation) a B-theory F, then it is a reasonable condition on a functor M that the composite MF is representable, in the sense of B. To be more precise, M is a model if MF is an exponential functor, and all that makes no reference to any ambient set theory. Lacking this definition, I don't see how the equivalence of theories and triples could possibly hold (my doubts in the second line). What am I missing here? Sincerely, Paul