Categories enriched on two sides

We introduce morphisms image of bicategories, more general than the original ones of Bénabou. When image, such a morphism is a category enriched in the bicategory image. 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 image to image in Caten induces a 2-functor image-image-Cat, while an adjunction between image and image in Caten induces one between the 2-categories image-Cat and image-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.