To try to clarify the remarks of David Yetter and Charles Wells on essentially algebraic theories, let me pass on an extract from the paper "Preframe Presentations Present" which I recently wrote with Peter Johstone (for "CT '90", to appear in Springer Lecture Notes in Mathematics). I ought to say that the section from which this extract is taken, which was intended to be a brief yet helpful account of essentially algebraic theories, is a distillation of Peter's knowledge rather than mine. "For a small essentially algebraic theory T, the forgetful functor from T-models to Set (or Set^n if T is many-sorted) has a left adjoint, just as in the algebraic case: the free T-model on a set X is constructed in the usual way as the set of words (i.e. terms) in the elements of X, modulo T-provable equality. The adjunction will not be monadic unless T is algebraic, but it will be possible to factor it as a tower of monadic adjunctions in the style of MacDonald and Stone ("The tower and regular decomposition", pp. 197-213 in Cahiers Top. Geom. Diff 23 (1982))." I presume, again without being an expert in these fields, that for finitary theories this all works over arbitrary elementary toposes with NNO. Steve Vickers