[{"_id":"some_uuid","inv":{"food01":"id01","food02":"id02","food03":"id03"},"food":{"id01":{"n":"apple"},"id02":{"n":"banana"},"id03":{"n":"pear"}}}]

db.collection.aggregate([{$set:{inv:{$objectToArray:"$inv"},food:{$objectToArray:"$food"},}},{$project:{inv:{$map:{input:"$inv",in:{k:"$$this.k",v:{$filter:{input:"$food",as:"food",cond:{$eq:["$$this.v","$$food.k"]}}}}}}}},{$project:{inv:{$map:{input:"$inv",in:{k:"$$this.k",v:{$first:"$$this.v.v.n"}}}}}},{$project:{inv:{$arrayToObject:"$inv"}}}])