February 3, 1995 To: Andrea Schalk, Andrea.Schalk@cl.cam.ac.uk From: Ernie Manes, manes@math.umass.edu Re: Your query yesterday on cat-dist The question you asked was solved in my 1967 thesis and appears as Exercise 9 on page 217 of my book on Algebraic Theories, Springer-Verlag GTM 28, 1976 where the reference to the thesis is also given. In effect, if T is your monad and C is the category of algebraic structures, you give yourself a functor J : F ---> C where F is the full subcategory of free T-algebras. This is because a typical homomorphism between free T-algebras has the form of a composition Tf followed by a mu. Your question then amounts to asking for a canonical extension of J to the category of all T-algebras. This construction, based on contractible coequalizers, is given in the references cited in the first paragraph. All the best, egm