The context of this article is the program to develop monoidal bicategories with a feedback operation as an algebra of processes, with applications to concurrency theory. The objective here is to study reachability, minimization and minimal realization in these bicategories. In this setting the automata are 1-cells in contrast with previous studies where they appeared as objects. As a consequence we are able to study the relation of minimization and minimal realization to serial composition of automata using (co)lax (co)monads. We are led to define suitable behaviour categories and prove minimal realization theorems which extend classical results. This work has been supported by NSERC Canada, Italian MURST and the Australian Research Council 1 Introduction Katis, Sabadini, Walters, and Weld have described bicategories equipped with operations of serial and parallel composition, and feedback modelled as, respectively, composition of 1cells, a tensor product and an operation called...