i have a problem about conditions in mongodb lookup

Question

Hellow freinds! i have a problem with aggregate in my project.

i have two collection that i want to select them by lookup. the first one is Cart:

const mongoose = require("mongoose")
const Schema = mongoose.Schema

const cartSchema = new Schema({
    product_id : {type : mongoose.ObjectId},
    user_cookie : {type : String},
    user_session : {type : String}
})

module.exports = mongoose.model("Cart", cartSchema, "Cart")

the second one is Product:

const mongoose = require("mongoose")
const Schema = mongoose.Schema

const productSchema = new Schema({
    name : {type : String},
    canonical : {type : String},
    no_index : {type : Boolean},
    category : {type : String},
    category_url : {type : String},
    description : {type : String},
    content : {type : String},
    image : {type : String},
    alt : {type : String},
    url : {type : String},
    meta_keywords : {type : String}
})

module.exports = mongoose.model("Product", productSchema, "Product")

the Cart.product_id is equal to Product._id. i also have a condition. the Cart.user_cookie should be equal to req.cookies.cookieName.

        const results = await Cart.aggregate([
            {
              $lookup: {
                from: "Product",
                let: { ids: ["$product_id"] },
                pipeline: [
                  {
                    $match: {
                      $expr: {
                        $in: ["$_id", "$$ids"] 
                      }
                    }
                  },
                  {
                    $match: {
                      $expr: {
                        $eq: ["$user_cookie", req.cookies.cookieName] 
                      }
                    }
                  }
                ],
                as: "joinedData",
              }
            },
            {
              $unwind: "$joinedData"
            }
          ]);

The output should be something like this:

results[
    {
        "_id": {
          "$oid": "65f484de010a337f71756933"
        },
        "product_id": {
          "$oid": "65f1d6b5a62a676f3a7b84af"
        },
        "user_cookie": "05230045185837191",
        "user_session": "",
        "name": "dddd",
        "canonical": "",
        "no_index": true,
        "description": "",
        "content": "",
        "image": null,
        "alt": "",
        "url": "",
        "meta_keywords": "",
        "__v": 0
      },
      {
        "_id": {
          "$oid": "65f4901d452327129855281b"
        },
        "product_id": {
          "$oid": "65f1d6b5a62a676f3a7b84af"
        },
        "user_cookie": "05230033185837191",
        "user_session": "",
        "name": "dddd",
        "canonical": "",
        "no_index": true,
        "description": "",
        "content": "",
        "image": null,
        "alt": "",
        "url": "",
        "meta_keywords": "",
        "__v": 0
      }
]

I would be very grateful if you could solve my problem.