For supplier work, keep the supplier relationship type, tier, contract control, fourth-party exposure, monitoring cadence, incident notice route, and exit evidence in one record.
ISO/IEC 27018 applies to public cloud services that process PII under contract to other organizations. In practice, the contract should state the subject-matter and duration of processing, the nature and purpose of processing, the type of personal data and categories of data subjects, and the controller's rights and obligations.
The processor should also be bound to documented controller instructions, confidentiality, appropriate technical and organizational security measures, controls on other processors, help with data subject requests and breach response, and deletion or return of data when the service ends.