Example: One-time maps do not determine the process

Process one SWAPs the input with an independent maximally mixed bath qubit and later SWAPs back. Process two applies a uniformly random Pauli, retains its classical label, and later applies the inverse. Without interventions both have a replacement channel at the first time and identity at the second. Replacing the carrier by $|0⟩⟨0|$ between the steps makes the first process return the original input and the second return $I_2/2$.