Dear Colleagues, I wonder whether the following description of a free F-conservative completion of any category under C-colimits (where F and C are classes of small categories such that F is a subclass of C and "F-conservative" means "preserving existing F-colimits"). The description of the cocompletion is as follows: Suppose C is a class of small categories and F is a subclass of C. Let X be any category. Denote by [X^op,Set] the quasicategory of all functors and all natural transformations between them. Denote by F^op-[X^op,Set] the quasicategory of all functors which preserve F^op-limits, i.e. limits of functors d : D -> X^op with D^op in F. Claim 1. F^op-[X^op,Set] is reflective in [X^op,Set] (The proof uses the fact that the above Claim holds for the case when X is small - Korollar 8.14 in Gabriel, Ulmer: Lokal pr"asentierbare Kategorien.) By Claim 1., F^op-[X^op,Set] has all small colimits. Denote by D(X) the closure of X (embedded by Yoneda) in F^op-[X^op,Set] under C-colimits. Then one can prove that D(X) is a legitimate category. The codomain-restriction I: X -> D(X) of the Yoneda embedding fulfills the following: 1. D(X) has C-colimits. 2. I preserves F-colimits. 3. D(X) has the following universal property: for any functor H : X -> Y which preserves F-colimits and the category Y has C-colimits there is a unique (up to an isomorphism) functor H* : D(X) -> Y such that H* preserves C-colimits and H*.I = H. In fact, this gives a 2-adjunction between C-CAT_C : the 2-quasicategory of all categories having C-colimits, all functors preserving C-colimits and all natural transformations and CAT_F : the 2-quasicategory of all categories, all functors preserving F-colimits and all natural transformations. The result also holds for V-categories, instead of a class C of small categories one has to work with a class of small indexing types. Thank you, Jiri Velebil velebil@math.feld.cvut.cz Department of Mathematics FEL CVUT Technicka 2 Praha 6 Czech Republic