The documentation https://docs.moodle.org/en/How_to_let_teachers_share_questions_between_courses says it all. Still here is a short report how we applied it.
Our obstacle was to understand that question categories do not propagate along course category hierarchies. The reason is clear: By default the teachers should not see each others' questions! If one tries to open this restriction by assigning the whole group the Teacher role in the Dept. course category, they all appear as teachers in every course of the Dept.
So the answer was to create the Question sharer role and assign this role to in their Dept. course category to all the teachers of the Dept.. In addition, the teacher who create shareable questions should move them up from their individual courses to the Dept. course category. That was all!
Looking for a simple method to share question categories
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