Does any body know the answer to the following question ? Is there a locally cartesian closed category C such that there is a full sub-left-exact category D of C which is non-reflective and forms an exponen- tial ideal, i.e. whenever A is an object in D and X is an object in C then [X->A] is (isomorphic to) an object in D ? The background to this question is the following. If something as described above would exist then one can perform the following construction. A morphism f : Y -> X in C is a DISPLAY MAP iff for any g : Z -> X : Pi (!Z) (g*f) : P -> 1 with P in D . For the collection DISP of display maps then one can prove : i) display maps are stable under arbitrary pullbacks ii) for any I in C : DISP / I ---> C / I reflects finite limits (iii) if f : X -> I is a display map and g : I -> J is an arbitary morphism in C then Pi (g) (f) is a display map as well . Now if D forms an exponential ideal then DISP / 1 is equivalent to D and we would get that the full subcategory of C (in the fibrational sense) is not reflective w.r.t. C / C . This would give an answer to a question I raised on this forum about one year ago. Thomas Streicher ======================================