We regard generalized twistor diagrams as linear SU(p, q)–equivariant operators on tensor products of ladder representations. We show the existence of cycles on which evaluation of the parallelcomposition of two box diagrams gives their operator product when it exists. Parallel composition involves tensor products with direct integral decomposition. Thus, the analytical situation is rather more difficult than in the sequential case where one has direct sums only. In contrast to the sequential case, our considerations here naturally include unbounded operators.