[{"part":[{"a":"1","b":"a"},{"a":"23","b":"b"},{"a":"4","b":"c"},]},{"part":[{"a":"1","b":"a"},{"a":"3","b":"b"},{"a":"5","b":"c"},]}]

db.collection.aggregate([{"$addFields":{joinStr:{$reduce:{input:"$part",initialValue:{a:"",b:""},in:{a:{$concat:["$$value.a","$$this.a"]},b:{$concat:["$$value.b","$$this.b"]}}}}}},{"$match":{"joinStr.a":"1234"}},{"$unset":"joinStr"}])