Proposition: Hybrid propagator and composition

Define the typed operational propagator by

With the event germs included as endpoints, every expression is a CPTP map between its declared interfaces and

for every compatible ordered triple.

Proof. The pre-event and post-event maps have their stipulated composition laws. A crossing interval contains exactly one readout. Splitting such an interval before the readout composes two pre-event factors; splitting it after the readout composes two post-event factors. In either case the factors reduce to the middle line of (eq:hybrid-T). No composition introduces a second ${\mathcal{M}}_C$. Complete positivity and trace preservation are preserved under composition.