Dear Stefan
[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?
Brian Day put out a short preprint: Brian J. Day, Biclosed bicategories: localisation of convolution, Macquarie Mathematics Reports #81-0030 (April 1981) but it was (allegedly) too far ahead of its time to be published. Here "biclosed" means that all right extensions and right liftings exist. So a one-object "biclosed bicategory" is a monoidal category with both left and right internal homs. In the short paper he defines what I think is exactly what you want and calls them "probicategories". By performing convolution on the homs one obtains biclosed bicategory which is locally cocomplete. In more recent work, Brian and I have found something more general than probicategories to be useful. Again, afraid of going too general, we have concentrated on the one object case; thus we have things called "substitudes" which are lax versions of promonoidal V-categories. They also generalise Lambek's multicategories. For the promonoidal case of a substitude, the multihoms are all determined up to canonical isomorphism by the nullary, unary and binary homs. See for example: 72. (with B.J. Day) Lax monoids, pseudo-operads, and convolution, in: "Diagrammatic Morphisms and Applications", Contemporary Mathematics 318 (AMS; ISBN 0-8218-2794-4; April 2003) 75-96. 77. (with B.J. Day) Abstract substitution in enriched categories, J. Pure Appl. Algebra 179 (2003) 49-63. The natural level of generality for the subject of 70. (with G.M. Kelly, A. Labella and V. Schmitt) Categories enriched on two sides, J. Pure Appl. Algebra 168 (1) (8 March 2002) 53-98 seems to be substitudes-with-several-objects rather than bicategories. I actually wrote some draft sections on that during the writing of [70] but we chickened out. Best wishes, Ross