This is simple. Use the following statement to return only the current matching array:

db.getCollection('web_mem_favorites').find({"_id":NumberLong(1181675746),"favorite_shards.sid":NumberLong(577)},{"favorite_shards.$":1}).pretty()

javascriptdb.getCollection('web_mem_favorites').find(
    {'_id':NumberLong(1181675746)},favorite_shards.sid': {'$in':[NumberLong(577)]}}
    {"_id" : 1, "favorite_shards": "$slice[2,1]" }
)

When the favorite_shards array is returned, only the second array element is returned.
But this requires knowing in advance which element sid:577 is.
In the mongodb array query manual, there is no method that can return array units that meet custom conditions. You can try to use a program to filter the favorites_shards data on the returned result set.

findOne( {'_id':NumberLong(1181675746),'favorite_shards.sid': {'$in':[NumberLong(577)]}} ,{"favorite_shards.$":1, "_id": 0} )

I understand the meaning of the question, here is the modified code

db.test.aggregate({"$unwind":"$favorite_shards"}, {"$match":{"favorite_shards.sid": {"$in": [NumberLong(578), NumberLong(577)]}}},  {"$group": {"_id": "$_id", "favorite_shards":{'$push': "$favorite_shards"}}})

Result:
{ "_id" : NumberLong(1181675746), "favorite_shards" : [ { "sid" : NumberLong(578), "favorite_dt" : ISODate("2015-06-26T0406.405Z"), " is_attention" : true }, { "sid" : NumberLong(577), "favorite_dt" : ISODate("2015-06-26T0548.449Z"), "is_attention" : true } ] }

You can use the projection operator $elemMatch:

javascriptdb.test.find({'favorite_shards.sid': 577}, {favorite_shards:{$elemMatch:{sid:577 }  } }).pretty()

{
    "_id" : NumberLong(1181675746),
    "favorite_shards" : [
        {
            "sid" : NumberLong(577),
            "favorite_dt" : ISODate("2015-06-26T05:20:48.449Z"),
            "is_attention" : true
        }
    ]
}

The limitation of $elemMatch is that it can only return the first matching record in the array.

I would like to ask what software this is?

You can use $unwind to query, and multiple subdocuments that meet the query conditions will be returned.

db.web_mem_favorites.aggregate(
    {"$project":{"favorite_shards":"$favorite_shards"}},
    {"$unwind":"$favorite_shards"},
    {"$match":{"favorite_shards.sid": NumberLong(577)}}
)