It sounds like you want to put a teacher tab on a node page of type department.

Create a view that shows content (nodes). Filter it to only show teachers. Add a relationship to the department.

Add a contextual filter of nid, and set the relationship (to department) you created on that contextual filter. Hide the view when the filter value is not available, and add validation to it that confirms that it is a nid belonging to a department.

Add a path to the view: node/%/teachers. Add a menu entry. Set it to tab, and give it a title of 'Teachers'.

It sounds like you want to put a teacher tab on a node page of type department.

Create a view that shows content (nodes). Filter it to only show teachers. Add a relationship to the department.

Add a contextual filter of nid, and set the relationship (to department) you created on that contextual filter. Hide the view when the filter value is not available, and add validation to it that confirms that it is a node nid belonging to a department.

Add a path to the view: node/%/teachers. Add a menu entry. Set it to menu tab, and give it a title of 'Teachers'.