Monads and limits in bicategories of circuits

We study monads in the (pseudo-)double category $\mathbf{KSW}(\mathcal{K})$ where loose arrows are Mealy automata valued in an ambient monoidal category $\mathcal{K}$, and the category of tight arrows is $\mathcal{K}$. Such monads turn out to be elegantly described through instances of semifree bicrossed products (bicrossed products of monoids, in the sense of Zappa-Sz\'ep-Takeuchi, where one factor is a free monoid). This result which gives an explicit description of the `free monad' double left adjoint to the forgetful functor. (Loose) monad maps are interesting as well, and relate to already known structures in automata theory. In parallel, we outline what double co/limits exist in $\mathbf{KSW}(\mathcal{K})$ and express in a synthetic language, based on double category theory, the bicategorical features of Katis-Sabadini-Walters `bicategory of circuits'.