Categories enriched on two sides

We introduce morphisms V --> W of bicategories, more general than the original ones of Benabou. When V = 1, such a morphism is a category enriched in the bicategory W. Therefore, these morphisms can be regarded as categories enriched in bicategories "on two sides". There is a composition of such enriched categories, leading to a tricategory Caten of a simple kind whose objects are bicategories. It follows that a morphism from V to W in Caten induces a 2-functor V-Cat -->W -Cat, while an adjunction between V and W in Caten induces one between the 2-categories V-Cat and W-Cat. Left adjoints in Caten are necessarily homomorphisins in the sense of Benabou, while right adjoints are not. Convolution appears as the internal horn for a monoidal structure on Caten. The 2-cells of Caten are functors; modules can also be defined, and we examine the structures associated with them. (C) 2002 Elsevier Science B.V. All rights reserved.