Corollary: Action on realizable finite operational objects

On full-tail objects with common logical interfaces, define

Within the explicit source–target class of this paper, Theorem thm:diagonal supplies full-tail realizations of these transformed objects and preserves witnessed matched modifications. The transformations satisfy the group law on these limit data.

Proof. Realizability and modifications follow from Theorem thm:diagonal. On geometric data, $g(hK)=(gh)K$ and similarly for acquisitions. Logical channels are unchanged under the fixed port identifications, giving the group law.