Hello, In the following reference [1] B.J. Day, On closed categories of functors, Lecture Notes in Math 137 (Springer, 1970) 1-38 are defined promonoidal, or monoidal enriched categories. It seems that there should be some well known many object version of this, in the sense that a bicategory is the many object version of a monoidal category. Does anyone know a definition or, even better, a reference? A much later related definition is in the appendix of [2] V. Lyubashenko, Category of $A_{\infty}$--categories, Homology, Homotopy and Applications 5(1) (2003), 1-48. Here are defined enriched 2-categories. This seems to be the strict case of what I'm looking for, since a promonoidal category is a monoid in the category of enriched categories, or a one-object category enriched over V-Cat. In [2] enriched 2-categories are defined as enriched over V-Cat. Thanks, Stefan Forcey