Categories enriched on two sides

We introduce morphisms V → W of bicategories, more general than the original ones of Bénabou. 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 homomorphisms in the sense of Bénabou, while right adjoints are not. Convolution appears as the internal hom 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.