The middle manager or Director is responsible for representing his department internally and externally. Consider each of the triangles as hierarchy of different functions such as HR, sales, product management, etc. Director is the first level interface between various departments. For instance Director of Engineering will talk to Director of HR for recruitment & training, will talk to Director of Marketing for getting the MRD in time, etc.
Each of these Directors has the specific goals assigned to them by their management in their triangles plus the organizational goal that all of them share.
The MM participates in cross-functional meetings, evaluates the supply chain internally and is always aware of the processes and context diagram. ie… where his inputs are coming, what are his dependencies, what are his outputs and who it should be fed to. The Directors discusses and negotiates on the schedule and services among them during the peer or cross-functional meeting and they communicate the goals and processes to their organization both up and down.
They bring the status updates regularly from their team, and update to their peers in other departments apart from their management. The Directors also have the responsibility to oversee the implementation of the processes and to audit periodically to check if its controls and systems are working.