Got some time to
See my answer to equivalent problem at Mongo query assistance update array element with field from object - #5 by steevej
Your problem is even simpler as you do not conditionally map. The following $set and $map should be good:
[ { "$set" :
{ "field_name.nested_array_of_obj" :
{ "$map" :
{ "input" : "$field_name.nested_array_of_obj" ,
"as" : "item" ,
"in" : { "field" : [ "$$item.field ] }
}
}
}
}
]