How to update a specific Array item by index, and copy the value of another field, in MongoDB?

I have many records in a MongoDB database, like this:

[
  {
    rootName: "AAA",
    outerItem: {
      innerItems: [
        {
          name: "A"
        },
        {
          name: null
        },
        {
          name: "C"
        }
      ]
    }
  }
]

I would like to copy the rootName field of each record into an outerItem.innerItems[1].newField only, using a named index position of 1 (or looking up the correct index in some other way).

What I expect is for each record to look like this, after the code is run:

[
  {
    "rootName": "AAA",
    "outerItem": {
      "innerItems": [
        { // 0
          "name": "A",
        },
        { // 1
          "name": null,
          "newField": "AAA" // New item should be added here only (array index 1)
        },
        { // 2
          "name": "C",
        }
      ]
    }
  }
]

I've tried using this code:

db.collection.update({},
[
  {
    "$set": {
      "outerItem.innerItems": {
        $map: {
          input: "$outerItem.innerItems",
          in: {
            $mergeObjects: [
              "$$this",
              {
                newField: "$rootName"
              }
            ]
          }
        }
      }
    }
  }
],
{
  "multi": false,
  "upsert": false
})

MongoDB Playground Link: https://mongoplayground.net/p/QhN1qAbWFwZ

But I end up with this:

[
  {
    "_id": ObjectId("5a934e000102030405000000"),
    "outerItem": {
      "innerItems": [
        {
          "name": "A",
          "newField": "AAA"
        },
        {
          "name": null,
          "newField": "AAA"
        },
        {
          "name": "C",
          "newField": "AAA"
        }
      ]
    },
    "rootName": "AAA"
  }
]

The problem is that the newField is copied to all the array items, when I only want it copied to 1 specific array item of index 1.

I tried to modify the code to use a condition of array index 1, but the following code does not work. It produces an error:

db.collection.update({},
[
  {
    "$set": {
      "outerItem.innerItems": {
        $map: {
          input: "$outerItem.innerItems",
          in: {
            $cond: [
              {
                $eq: [
                  "$$item.id",
                  1
                ]
              }
            ],
            $mergeObjects: [
              "$$this",
              {
                newField: "$rootName"
              }
            ]
          }
        }
      }
    }
  }
],
{
  "multi": false,
  "upsert": false
})

MongoDB Playground link: https://mongoplayground.net/p/ckgDCUN1XrR

This version also doesn't work:

db.collection.update({
  "outerItem.innerItems": {
    $elemMatch: {
      "name": null
    }
  }
},
[ ... ]

I can't figure out how to get it working. I have used the following guides so far, but I still cannot find a working solution for this scenario.

• Mongo updateMany statement with an inner array of objects to manipulate
• update one element of array of objects in mongodb
• Update array at specific index by other filed in MongoDB -- this example uses operators like $concatArrays with a $cond (condition) on the array
• https://www.mongodb.com/docs/manual/reference/method/db.collection.updateMany/
• MongoDB: How do I update a single subelement in an array, referenced by the index within the array?
• https://www.mongodb.com/community/forums/t/how-can-we-assign-one-field-value-to-another/16396/4
• https://www.geeksforgeeks.org/how-to-update-mongodb-field-using-value-of-another-field/

How do I get it working? The final code is intended to be used in Node.js TypeScript code, probably using something like:

await queryRunner.updateMany('name_items', {}, [ { $set: ... } ] );