Hi, given a Set-endofunctor F:Set-->Set and a "type" L of limit I'm searching for a "related" functor F_L:Set--->Set that preserves that type of limit. Example: As Vera Trnkova has proved around 1970 each Set-endofunctor can turned into a functor that preserves pullbacks of injective mappings just by modifying it on the empty set and the empty mappings. - Now given a functor that does not preserve pullbacks I'd like to find a "related" one that does. I'm not quite sure what "related" means. My first idea was to consider the functor category Fun(Set,Set) (not being abhorred by possible size problems) and the subcategory Fun_L(Set,Set) of Set-endofunctors that preserve limits of type L. Then if Fun_L(Set,Set) were reflective or coreflective in Fun(Set,Set) ... but I think it is not. So does anybody have an idea which type of "relatedness" could be considered? Is there some concept concerning these problem in category theory? Thank you very much Tobias Schroeder -------------------------------------------------------------- Tobias Schröder FB Mathematik und Informatik Philipps-Universität Marburg WWW: http://www.mathematik.uni-marburg.de/~tschroed email: tschroed@mathematik.uni-marburg.de