Reply to Peter Freyd's answer to my question about full internally complete sub- lccc-s : the example you have given is a perfectly good answer to the question for the nonfibred or noninternal case. What you have actually done is to give a lccc together with a full subcategory which is not reflective but small complete when looked at it externally from the point of view of the category of sets. Actually Peter Johnstone has constructed a similar example, namely On + On* where On is the linear order of all ordinals and On* is the opposite categor The nonreflective full subcategory which is small complete is of course On*. Your example is nearly the same but even a little bit more economical by choos- ing the one element category instead of On. Now why am I not quite satisfied. You have given a perfectly good answer to my second (easier!) question, but not to my first and more difficult question. In your mail you said : "By every definition I can think of it is internally complete." I show a definition of internal completeness w.r.t. which the subcategory you have shown is not NOT internally complete ! Let C be a lccc. Then a full subcategory of C is given by a class D of dis- play maps, i.e. some class D of C-morphisms which is stable under pullbacks along arbitrary C-morphisms. That means C gives a full subfibration of the fibration cod : C^2 -> C . Now by "internally complete" I mean that the full subfibration as given by D is complete in the usual sense as defined by J. Benabou. Expressed in an elementary way that means that the following three conditions are fulfilled : (1) any identity morphism is in D (internal terminal objects) (2) if f : B -> A , g : C-> A are arbitrary C-morphisms and p : P -> B , q : P -> C is the pullback cone for f and g in C then if a : A -|> I is display map such that a o f and a o g are display maps then a o f o p (= a o g o q) is a display map, too (internal pullbacks) and finally a condition expressing that the full subfibration has internal products (3) if f : B -> A is an arbitrary C-morphism and a : C -|> B is a display map 'Pi'f(g) is a display map, too (where 'Pi'f is the right adjoint to f*, the pullback functor in C) . Now what I want to have is a lccc C and a class D of display maps such that the conditions (1),(2),(3) are all satisfied and the cartesian embedding of the subfibration cod : D^2 -> C into cod : C^2 -> C has a CARTESIAN left adjoint. Now what is the reason that any lccc which is a linear order cannot serve as an example of the kind I want. Linear orders with a greatest element although surely lccc-s are highly degener- ate in the sense that left and right adjoints to pullback functors coincide (!!) Thus as soon as I have a class of display maps satisfying conditions (1) and (3) I can prove that any arbitrary morphism is a display map : let f : B -> A be an arbitrary morphism, by (1) the identity morphism id(B) : B -|> B is a display map by (3) we have that 'Pi'f(id(B)) is a display map, too BUT as 'Pi'f(id(B)) = 'Sigma'f(id(B)) = f o id(B) o f we know that f itself is a display map. So one can see that the constraint of internal completeness is much more stronge r than the constraint of small completeness. Perhaps one could formulate the question in terms of 'concrete categories'. Does it hold that for any full concrete category having small limits which are inherited from Set that it is already a full reflective subcategory of Set itsel f ?? If this would hold (and I don't know!) than one could try to show that this eventual theorem can be internalised in the sense that instead of working with Set work relative to an arbitrary lccc C a display maps REPRESENT full concrete fibred categories. Finally I would say that I am not so sure at all whether such an example can be found at all. I would appreciate a proof that is is impossible to find such an example even more. Some indication is given by the following fact. In Set the only full reflective subcategories closed under binary products, exponentiation and equlaisers are isomorphic to { {}, {{}} } . Be careful with "closed under exponentiation I mean that if A is in the subcategory and X is arbitrary then X => A is in the sub- category ! Ok, in the category of 'w'-sets this is not quite true because of the full sub- category of per-s. There is even an internally complete full subcategory of 'w'- Set which is not small but nevertheless reflective. It is obtained by taking the FREYD COVER of the 'w'-Set model. There propositions are sums of families of set s indexed over pers. This can be easily seen to be large, but nevertheless refle ctive. Perhaps one can show the following weak version of what holds in Set. Any full internally complete subcategory of an lccc in the fibrational sense is already reflective. In Set it is not only reflective but highly trivial, namely a (pre-)order. Proving this would be a considerable strengthening of FAFT in a special case !! Thomas Streicher