Find price for every ingredient in product and calculate price

I have a list of products mapped in a table in react and i can edit them adding ingredients and quantity of them. I need to be able to find every instance of ingredient in every product, look up price for every ingredient in other document, multiply it by quantity, for example:

1. "Soup" has sub array called "ingredients", one fo them is "milk" with quantity of 0,3. It looks up for "milk" in "Ingredients" document, finds price for milk, and multiples it - 5 * 0,3 (for every ingredient in product) and gives back result inside product "price"

Products:

{
  "_id": {
    "$oid": "62e391b2a16efedc01d71e35"
  },
  "name": "Soup",
  "ingredients": [
    {
      "name": "milk",
      "quantity": "0,3",
      "_id": {
        "$oid": "62e391b2a16efedc01d71e36"
      }
    },
    {
      "name": "flakes",
      "value": "0,2",
      "_id": {
        "$oid": "62e391b2a16efedc01d71e37"
      }
    }
  ],
  "__v": 2
}

Ingredients:

{
  "_id": {
    "$oid": "6296689c4a87b691e7e04d39"
  },
  "name": "milk",
  "__v": 0,
  "price": 5
},
{
  "_id": {
    "$oid": "6296689c4a87b691e7e04d39"
  },
  "name": "flakes",
  "__v": 0,
  "price": 2
}

Expected result:

{
  "_id": {
    "$oid": "62e391b2a16efedc01d71e35"
  },
  "name": "Soup",
  "price": 1,9,
  "ingredients": [
    {
      "name": "milk",
      "quantity": 0,3,
      "_id": {
        "$oid": "62e391b2a16efedc01d71e36"
      }
    },
    {
      "name": "flakes",
      "quantity": 0,2,
      "_id": {
        "$oid": "62e391b2a16efedc01d71e37"
      }
    }
  ],
  "__v": 2
}

EDIT: I tried with this but it's only adding first item of array outside of subarray of ingredients and i have no idea how to use it for every item:

Products.aggregate(
    [
      {
        $lookup: {
          from: "ingredients",
          localField: "ingredients.name",
          foreignField: "name",
          as: "ingredients.price",
        },
      },
      {
        $set: {
          price: { $arrayElemAt: ["$ingredients.price", 0] },
        },
      },
    ],
    function (err, ingredients) {
      if (err) throw err
      console.log(JSON.stringify(ingredients, null, 4))
    }
  )