Getting Values from Nested Mongo DBRefs - mongodb

Question

I'm having an issue with retrieving DBRef values present in an array of sub documents. I've had success using $lookup to retrieve the data from a top level DBRef, but can't seem to put it together when the DBRef is nested inside an array.
In this situation, I have a shipment document that looks like this:
{
id: '123',
title: 'Example'
shipmentLotDetails: [
{
lot: DBRef("inventoryLot", ObjectId("12345")),
quantity: 10
},
{
lot: DBRef("inventoryLot", ObjectId("16587")),
quantity: 5
}
]
}
The lot document looks like this:
{
id: '12345',
title: 'Another Example',
cost: 1.50
}
My goal is to retrieve the cost value for each element in shipmentLotDetails so that I can calculate the cost of units in a shipment. That calculation would just be lot.cost * quantity.
I found a post that describes a process similar to what I'm trying to do here: Mongo Lookup with nested DBRefs. However, the solution process is not explained well enough for me to follow and adapt to my situation.
I know that I need to do a $map on shipmentLotDetails and believe it should look something like this:
db.shipment.aggregate([
{
$project: {
shipmentCost: {
$map: {
input: "$shipmentLotDetails",
in: { $arrayElemAt: [{ $objectToArray: "$$this" }, 0]}
}
}
}
}
])
I know I would also need to do a $lookup to actually pull the lot information from the DBRef, but am not sure where to place that.
Any direction on this would be greatly appreciated. Thank you!