如何修改 mongodb 中嵌套对象数组中的字段名称键

我有一个MongoDB集合，其中有许多这样的对象:

{
 "_id" : "1234", 
 "type" : "automatic", 
 "subtypes" : [
    {
     "_id" : "dfgd",
     "name" : "test subtype", 
     "subjetRequired" : true,
    },
    {
     "_id" : "dfgd",
     "name" : "test subtype2", 
     "subjetRequired" : false,
    }
  ],
 "anotherField" : "some value"
}

As you can see, one of the keys in the subtypes array is incorrectly spelled - "subjetRequired" instead of "subjectRequired".
I want to correct that key name. How can I do that.
I'll preface this by saying I've not worked with mongodb very much in the past.
After a lot of researching, the best I could come up with is the following (which doesn't work):

function remap(doc) {
     subtypes = doc.subtypes;
     var count = 0;
     subtypes.forEach(function(subtype){
         db.taskType.update({"_id": subtype._id}, {
             $set: {"subtypes.subjectRequired" : subtype.subjetRequired},
             $unset: {"subtypes.subjetRequired": 1}
         });
     }
     )
}

db.taskType.find({"subtypes.subjetRequired":{$ne:null}}).forEach(remap);

This doesn't work.
I know the loop is correct, as if I replace the other logic with print statements I can access and print the fields who's names I want to modify.
What am I doing wrong here?