How to pull an array element from array nested inside an object in mongoose?

I am trying to remove the product from the users cart when the product is deleted.

My User Schema:

const userSchema = new Schema({
    username: { type: String, required: true },
    phone: {
        type: String,
        match: [/\d{10}/, 'Phone number should only 10 have digits'],
    },
    email: {
        type: String,
        match: [/\S+@\S+\.\S+/, 'Email must be valid'],
        required: [true, 'Email must be valid'],
        unique: true,
    },
    password: {
        type: String,
        required: [true, 'Password is not valid'],
    },
    cart: {
        items: [
            {
                productId: {
                    type: Schema.Types.ObjectId,
                    ref: 'Product',
                },
                quantity: {
                    type: Number,
                    max: [10, 'Only 10 quantity of one item can be added.'],
                    required: true,
                },
            },
        ],
        total: { type: Number, default: 0 },
    },
    admin: { type: Boolean, default: false },
    addresses: [addressSchema],
});

I tried this code to remove the item from items array:

await User.updateMany({}, { cart: { $pull: { 'items.productId': id } } });

However this deletes the whole cart object instead of single items. Please someone help me with this.

Take a look at this official example Remove All Items That Equal a Specified Value

{ $pull: { <field1>: <value|condition>, <field2>: <value|condition>, ... } }

The field1 should be an array, cart.items is an array. This means we pull some elements from the cart.items array.

db.collection.updateMany({},
{
  $pull: {
    "cart.items": {
      productId: 2
    }
  }
})

Input:

[
  {
    "_id": ObjectId("5a934e000102030405000000"),
    cart: {
      items: [
        {
          productId: 1
        },
        {
          productId: 2
        },
        {
          productId: 3
        },
        
      ]
    }
  }
]

Output:

[
  {
    "_id": ObjectId("5a934e000102030405000000"),
    "cart": {
      "items": [
        {
          "productId": 1
        },
        {
          "productId": 3
        }
      ]
    }
  }
]

mongoplayground