Database 位置运算符($)工作不正确

我正在try 从购物车集合的物品数组中查找(并更新)特定对象. 购物车文档看起来像这样

{
    _id: ObjectId("661fb9ddc2b236ae8703ba92"),
    display_id: '38bd6126-02a1-47b9-b567-00aaf2eb076f',
    user_id: ObjectId("661137f22a31832ceb92ddbc"),
    items: [
      {
        product_id: ObjectId("6622376051d5886f39ac7365"),
        size: ObjectId("660d444d0d7716ee24129675"),
        qty: 3
      },
      {
        product_id: ObjectId("662214f8cc5306f999369e99"),
        size: ObjectId("660d444d0d7716ee24129675"),
        qty: 2
      },
      {
        product_id: ObjectId("662214f8cc5306f999369e99"),
        size: ObjectId("660d444d0d7716ee24129674"),
        qty: 2
      }
    ],
    __v: 4
  }

我try 运行的查询是

db.carts.find({user_id: new ObjectId("661137f22a31832ceb92ddbc"), "items.product_id": new ObjectId("662214f8cc5306f999369e99"), "items.size":new ObjectId("660d444d0d7716ee24129675")}, {"items.$":1})

或者是mongoose

const doc = await Cart.findOne({
                user_id: user_id,
                "items.product_id": product_id,
                "items.size": oldSize
            },{"items.$":1})

理想情况下它应该返回这个对象:

{
 product_id: ObjectId("662214f8cc5306f999369e99"),
 size: ObjectId("660d444d0d7716ee24129675"),
 qty: 2
}

但它返回这个:

{
 product_id: ObjectId("6622376051d5886f39ac7365"),
 size: ObjectId("660d444d0d7716ee24129675"),
 qty: 3
},

请告诉我我做错了什么.