  We extend the free cornering of a symmetric monoidal category, a double
categorical model of concurrent interaction, to support branching communication
protocols and iterated communication protocols. We validate our constructions
by showing that they inherit significant categorical structure from the free
cornering, including that they form monoidal double categories. We also
establish some elementary properties of the novel structure they contain.
Further, we give a model of the free cornering in terms of strong functors and
strong natural transformations, inspired by the literature on computational
effects.
