In answer to Pat Donaly The notion that Pat proposes has been defined under the name of "U-dominated category" (categorie U-dominee) par Charles Ehresmann in the early sixties (before the notion of an enriched category was introduced), and he has extensively used it in several of his papers during these years ; later he rallied the stricter notion of enrichment. Most of these papers are re-printed in. "Charles Ehresmann : Oeuvres completes et commentees" Part III-2. The definition, introduced in 1963 (cf. p. 492), is also briefly recalled in his book "Categories et Structures" (Dunod 1965, p. 81). An interesting application is given in the (not easy to read) paper "Cohomologie a valeurs dans une categorie dominee" (1966) which contains a lot of ideas which have not been developed later on but would certainly lead to interesting results. In this same volume of the "Oeuvres" I have given a comparison between dominated categories and enriched categories (Comment 699.1 page 829). Andree C. Ehresmann