From coherent structures to universal properties

Given a 2-category image admitting a calculus of bimodules, and a 2-monad image on it compatible with such calculus, we construct a 2-category image with a 2-monad image on it such that • image has the adjoint-pseudo-algebra property. • The 2-categories of pseudo-algebras of image and image are equivalent. Thus, coherent structures (pseudo-image-algebras) are transformed into universally characterised ones (adjoint-pseudo-image-algebras). The 2-category image consists of lax algebras for the pseudo-monad induced by image on the bicategory of bimodules of image. We give an intrinsic characterisation of pseudo-image-algebras in terms of representability. Two major consequences of the above transformation are the classifications of lax and strong morphisms, with the attendant coherence result for pseudo-algebras. We apply the theory in the context of internal categories and examine monoidal and monoidal globular categories (including their monoid classifiers) as well as pseudo-functors into image.