Dear all, I've recently been trying to understand the notion of "the" free category with X over a given category. Here X is supposed to be something like "finite limits" or "natural number object" or "cartesian closed structure" or combinations of similar properties. My interest in this was fueled by reading sentences such as (if I understand correctly): "X is algebraic structure on Cat, so there's a monad for adding X and so the categories with X are the algebras for that monad". Unfortunately there seem to be few detailed and explicit examples of monads on Cat in the standard books. Clearly one has to be careful with regard to what should be termed "the free category". For example, in the case of just adding products we would have to decide whether or not we wanted a x (b x c) and (a x b) x c to be equal or isomorphic or even (perhaps?) isomorphic up to some particular isomorphism. But suppose we've decided that. Then there is still the question of which morphisms are involved in each construction. Do our functors preserve the structure in the usual up-to-iso sense or strictly ("on the nose")? This is an important point since, for example, if P the the category of categories-with-products and functors preserving them up-to-iso, then it's not hard to show that the forgetful U : P --> Cat doesn't have a left adjoint. On these grounds one could perhaps argue that there isn't a free category with products over a given category. Of course, there are constructions which give what could reasonably lay claim to being "a" free such category. (Lambek and Scott's book (p.55) gives one such construction, but their functors are required to preserve structure strictly.) There is also the question of the free category with X preserving existing X over a given category and to what extent this can be done. (For example, Lambek and Scott's construction would destroy any products that happened to exist originally.) What I'm looking for therefore is references to the main papers / reports (or maybe even someone has some lecture notes?) in which details of these kinds of constructions are given or pointers to theorems by which I might be able to work out when certain free categories (preserving certain structures or otherwise) exist and when they don't. All help would be very gratefully appreciated. Many thanks, Don MacInnes