In the Task Details page there is a drop down box where a resource can select a Task Supervisor's name to indicate the person from their company who will be carrying out that task.
To add a name to this list you will need to go into the admin area of the application and add this person to your list of internal organisations.
Click here to view a step by step guide on Create an Organisation > to add a supervisors name go to the section; 'Individual, Employee, Manager or Contractor'.