Bicategories of Automata, Automata in Bicategories

We study bicategories of (deterministic) automata, drawing from prior work of Katis-Sabadini-Walters, and Di Lavore-Gianola-Rom\'an-Sabadini-Soboci\'nski, and linking their bicategories of `processes' to a bicategory of Mealy machines constructed in 1974 by R. Guitart. We make clear the sense in which Guitart's bicategory retains information about automata, proving that Mealy machines \'a la Guitart identify to certain Mealy machines \'a la K-S-W that we call fugal automata; there is a biadjunction between fugal automata and the bicategory of K-S-W. Then, we take seriously the motto that a monoidal category is just a one-object bicategory. We define categories of Mealy and Moore machines inside a bicategory B; we specialise this to various choices of B, like categories, relations, and profunctors. Interestingly enough, this approach gives a way to interpret the universal property of reachability as a Kan extension and leads to a new notion of 1- and 2-cell between Mealy and Moore automata, that we call intertwiners, related to the universal property of K-S-W bicategory.