Mongo DB:Como copiar o documento de uma coleção e adicioná-lo como um campo a um documento relacionado de outra coleção?

db.products.aggregate([       
   { /**
    * from: The target collection.
    * localField: The local join field.
    * foreignField: The target join field.
    * as: The name for the results.
    * pipeline: The pipeline to run on the joined collection.
    * let: Optional variables to use in the pipeline field stages.
    */
   $lookup: {
     from: 'events',
     localField: '_id',
     foreignField: 'product_id',
     as: 'events'
   }},
   {/**
    * into: The target collection.
    * on: Fields to  identify.
    * whenMatched: Action for matching docs.
    * whenNotMatched: Action for non-matching docs.
    */
   $merge: {
     into: 'products',
     on: "_id",
     whenMatched: 'merge',
     whenNotMatched: 'insert'
   }}
])

{ $match: { events: { $exists: false } }