I have the following question about factorisation of partial geometric morphisms (i.e. pullback preserving functors having right adjoints). It is well known that a pullback preserving functor F : A -> B between lex categories factors as A ---> B/F1 ---> B (where F_1 X = F(X->1) ) F_1 Sigma_F1 where, of course, F_1 is lex. Moreover F_1 has right adjoint eta_1* o U_F1 provided F -| U. Now my question is whether one can obtain something similar if A and B are PARTIAL LEX, i.e. have pullbacks and binary products but not a terminal object. In that case we can perform for every I in A the same factorisation as above A/I ---> B/FI ---> B (where F_I(x:X->I) = F(x) ) F_I Sigma_FI What I wonder is whether there exists a maximal factorisation of F as A -> C -> B F' S s.t. F' preserves pullbacks and binary products and S preserves pullbacks and has a right adjoint T and F' has a right adjoint if F has a right adjoint . I have a tentative answer to that : namely take for C the pseudo-colimit of B/F(-) : A -> Cat. Then everything works but the requirement that S has a right adjoint. I'd be grateful for any pointers to literature, folklore results or ideas. Thomas Streicher