In the previous work by Jacobs, Sokolova and the author, synchronous parallel composition of coalgebras--yielding a coalgebra--and parallel composition of behaviors--yielding a behavior, where behaviors are identified with states of the final coalgebra--were observed to form an instance of the microcosm principle. The microcosm principle, a term by Baez and Dolan, refers to the general phenomenon of nested algebraic structures such as a monoid in a monoidal category. Suitable organization of these two levels of parallel composition led to a general compositionality theorem: the behavior of the composed system relies only on the behaviors of its constituent parts. In the current paper this framework is extended so that it accommodates any process operator--not restricted to parallel composition--whose meaning is specified by means of GSOS rules. This generalizes Turi and Plotkin's bialgebraic modeling of GSOS, by allowing a process operator to act as a connector between components as coalgebras.