An interpretation of the Casselman-Wallach Theorem is that the K K -finite functor is an isomorphism of categories from the category of finitely generated, admissible smooth Fréchet modules of moderate growth to the category of Harish-Chandra modules for a real reductive group, G G (here K K is a maximal compact subgroup of G G ). In this paper we study the dependence of the inverse functor to the K K -finite functor on parameters. Our main result implies that holomorphic dependence implies holomorphic dependence. The work uses results from the excellent thesis of van der Noort. Also a remarkable family of universal Harish-Chandra modules, developed in this paper, plays a key role.