[{_id:ObjectId("5a934e000102030405000000"),"categories":[{name:"test",_id:ObjectId("5a934e000102030405000001"),products:["5a934e000102030405000001","5a934e000102030405000002"]},{name:"test2",_id:ObjectId("5a934e000102030405000003"),products:["5a934e000102030405000003"]}]}]

db.collection.update({_id:ObjectId("5a934e000102030405000000")},{$pull:{"categories.$[category].products":"5a934e000102030405000001"}},{arrayFilters:[{"category.products":"5a934e000102030405000001"}]})