Django: how to annotate queryset with count of filtered ForeignKey field?

Avoid extra and raw whenever possible. The aggregation docs have nearly this use case:

Straight from the docs:

# Each publisher, each with a count of books as a "num_books" attribute.
>>> from django.db.models import Count
>>> pubs = Publisher.objects.annotate(num_books=Count('book'))
>>> pubs
[<Publisher BaloneyPress>, <Publisher SalamiPress>, ...]
>>> pubs[0].num_books
73

So, to modify this for your particular example:

depts = Department.objects.
            filter(product__review__time__range=["2012-01-01", "2012-01-08"]).
            annotate(num_products=Count('product'))

The function calls on separate lines is just for readability and you should move them about accordingly. I haven’t tested this, but I think it should work.