Updating MongoDB document in embedded array with Mongoose, Node and Express

I'm trying to update a property in a MongoDB document that's in an embedded array. The JSON response shows that the field has been updated, but it's not storing that result in the database.

Here are my schemas;

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


let coinSchema = new mongoose.Schema({
  id: {
    type: String
  },
  name: {
    type: String
  },
  amount_owned: {
    type: Number,
    default: 0
  },
}, {
  timestamps: true
})

let Coin = mongoose.model('Coin', coinSchema)
module.exports = Coin;

let userSchema = new mongoose.Schema({
  name: {
    type: String
  },
  email: {
    type: String,
    required: true
  },
  list: []
})

let User = mongoose.model('User', userSchema);
module.exports = User;

and my controller update method;

function update(req, res, next) {
  User.findOne({
    _id: req.params.id
  }, function(err, user) {
    if (err) return console.log(err)
    for (let i = 0; i < user.list.length; i++) {
      if (user.list[i].id === req.body.coinId) {
        user.list[i].amount_owned = req.body.amount_owned
      }
    }
    user.update(function(err, user) {
      if (err) return console.log(err)
      res.json(user.list)
    })
  })
}

I'm using postman and specifying the updated 'amount_owned' field in the put request body, but I'm unable to save the result in the database, even though the JSON response shows that field has been updated.

via Aaron Goldsmith