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Dear Bob: Since Paul Johnson put his question on the full net, let me answer it there. 1. An equational theory is sketchable iff the theory has rank. This follows from theorems of Pare and Makkai and is sort of obvious anyway. 2. Yes, provided the base is accessible. 3. You had better ask Makkai or Pare. But my guess is the following: If you look at all the categories that are lambda-accessible, then there will be canonical sketches built from the lambda-accessible models. But there are theories of arbitrarily high arities whose models are, for example, the category of sets. Sounds paradoxical? Hint: the underlying functor is not the usual, but is the functor represented by a big (but still small) set. Michael