Thomas Streicher asks: I am interested in the following question. Given any lccc (locally cartesian closed category) C can one find a full internally complete sub-category which is NOT reflective ? ... I also would be quite satisfied with a counterexample in the non-fibred case. But I guess in the category of sets such a counterexample cannot be found. I reply: Are you sure this is the right question? Consider the dual of the order-type of the ordinal numbers. View it as a category and formally adjoin an intial object. This category is an lccc (as is any linearly ordered set when viewed as a category). The full subcategory obtained by removing the initial object is not reflective: there is no reflection for the initial object. It is in the standard sense a complete category. By every definition I can think of it is internally complete In case one is unhappy with this example because the subcategory is only complete not co-complete, modify the example by adjoining to the ambient category (and the subcategory) a new formal initial ofject. The previous initial object still lacks a reflection. Peter Freyd