  We introduce the normal produoidal category of monoidal contexts over an
arbitrary monoidal category. In the same sense that a monoidal morphism
represents a process, a monoidal context represents an incomplete process: a
piece of a decomposition, possibly containing missing parts. We characterize
monoidal contexts in terms of universal properties. In particular, symmetric
monoidal contexts coincide with monoidal lenses, endowing them with a novel
universal property. We apply this algebraic structure to the analysis of
multi-party interaction protocols in arbitrary theories of processes.
