I found: it was “Selected Objet”, as the relation to the menu item IS an object. The value or the tag will of course be correct, because they belong to the object.
More, I made everything work by adding a new relationship between “OBJ” and “ACTION”.
If I don’t, the inverse relationship always refers to the same (self) object. So, by changing a relationship, I was changing the object itself!
By adding a new one-to-many relationship, I can indirectly make a OBJ to OBJ relation (the targetObject and conditionObject may be the same one) AND keep the graph consistent.
Each day, I understand better what “Core Data is not for beginners” means.