The comparison function is part of the type so you will have to change the map type everywhere you use it. To make it easier to change the type in the future you can create a typedef.
typedef map<string,list,functor> MyMapType;
Now you can use MyMapType everywhere instead of map<string,list,functor>. If you later want to change the type again you only need to change one line of code.
This is just for demonstration. You probably want a better descriptive name for your typedef than I used here.