Ref.: The last message of M. Barr on regepis, etc..., and the one of P. Taylor The claim of Joyal is probably the one mentioned on p.292 of [Cassidy-Hebert- Kelly; J.Austr. Mat.Soc.38 (1985)], where it is assumed that (E,M) = (Strong Epis, Monos) (it is Kelly who reported that). Certainly it is not true for all E: to be more precise, Th. 3.3 and 4.7 of the paper imply that factorization systems (E,M) with E stable under pullbacks (in a finitely complete and well- powered category A) are exactly those couples (E,E*) with E = { f ; r(f) iso }for some left-exact reflection functor r from A and E* obtained by the diagonalproperty (Hence in particular all localisation morphisms a-->r(a) are in such an E, and classical cases give counterexamples). About a previous message about slices, it seems to me that the existence of a multi-initial object (in the sense of Diers) does not imply the existence of an initial object in each slice, unless you assume the existence of equalizers (A counterexample being the algebraically closed fields).(Am I using a different definition of multi-initial object?).In fact, at least for axiomatic categories of finitary models, the existence and preservation of equalizers by the forgetful functor is precisely what makes the difference between multi- adjointness and "local adjointness" (in the sense of [Kaput] and [Adamek- Volger] mentioned by Tholen). About preservation of pullbacks only, note that in axiomatic categories of finitary models, the existence and preservation by the forg.funct. of pullbacks imply the one of wide pullbacks (and filtered colimits) (This follows from results of [Pare, Can. J. Math 42 (1990), 731-746] and Volger: see [Volger, Initial and quasi-initial models of theories, manuscript, 1991]). Doesn't this permit to simplify the definition of a locally-finitely poly-presentable category (as an accessible category with pullbacks)? =====================================================================