  The bicategorical point of view provides a natural setting for many concepts
in the representation theory of monoidal categories. We show that centers of
twisted bimodule categories correspond to categories of 2-dimensional natural
transformations and modifications between the deloopings of the twisting
functors. We also show that dualities lift to centers of twisted bimodule
categories. Inspired by the notion of (pre)bimonoidal functors due to McCurdy
and Street and by bilax functors of Aguiar and Mahajan, we study 2-dimensional
functors which are simultaneously lax and colax with a compatibility condition.
Our approach uses a sort of 2-categorical Yang-Baxter operators, but the idea
could equally be carried out using a kind of 2-categorical braidings. We show
how this concept, which we call bilax functors, generalize many known notions
from the theory of Hopf algebras. We propose a 2-category of bilax functors
whose 1-cells generalize the notions of Yetter-Drinfel`d modules in ordinary
categories, and a type of bimonads and mixed distributive laws in 2-categories.
We show that the 2-category of bilax functors from the trivial 2-category is
isomorphic to the 2-category of bimonads, and that there is a faithful
2-functor from the latter to the 2-category of mixed distributive laws of Power
and Watanabe.
