This is in reply to Paul Johnson's question about constructing free categories over graphs in elementary topoi (with natural numbers object). Your own work on the indexed adjoint functor theorem is relevant to this(see note from Bob R. below), as are the following: B. Lesaffre, Structures algebriques dans les topos elementaires, C.R. Acad. Sci. Paris, 277 (1973) A663-666. D. Schumacher, Absolutely free algebras in a topos containing an infinite object, Canad. Math. Bull. 19 (1976) 323-328. Later, I wrote a paper (Words, free algebras, and coequalizers, Fund. Math. 117 (1983)117-160) showing that the usual set-theoretic existence proof works in the internal logic of topoi. (The emphasis there was on varieties with infinitary operations, and things can be simplified in the finitary case.) Andreas Blass ++++++++++++++++++++++++ The article of mine Andreas refers to is: On algebras defined by operations and equations in a topos, JPAA 17(1980) 203-221 but David's reply answers the question completely ... Bob Rosebrugh ========================================