Aprendí MongoDB durante los últimos dos días y estoy tratando de agregar tres colecciones pero no pude lograrlo

A continuación se muestran las cuatro colecciones que se mantienen en la base de datos.

Universidad

{
   "_id" : "5834ecf7432d92675bde9d82",
   "name": "NIFT"
}

Universidad

{
   "_id" : "5834ecf7432d92675bde9d83",
   "name": "NIFT Hyderabad",
   "university_id":"5834ecf7432d92675bde9d82"
}

departamentos

{
   "_id" : "5834ecf7432d92675bde9d84",
   "department_name": "Fashion Technology",
   "college_id" : "5834ecf7432d92675bde9d83" 
},
{
   "_id" : "5834ecf7432d92675bde9d85",
   "department_name": "Merchandising",
   "college_id" : "5834ecf7432d92675bde9d83"
}

Secciones

{
   "_id" : "5834ecf7432d92675bde9d86",
   "section_name": "A",
   "students" : "56",
   "department_id":"5834ecf7432d92675bde9d84"
},
{
   "_id" : "5834ecf7432d92675bde9d87",
   "section_name": "B",
   "students" : "60",
   "department_id":"5834ecf7432d92675bde9d84"
},
{
   "_id" : "5834ecf7432d92675bde9d86",
   "section_name": "A",
   "students" : "55",
   "department_id":"5834ecf7432d92675bde9d85"
},
{
   "_id" : "5834ecf7432d92675bde9d87",
   "section_name": "B",
   "students" : "44",
   "department_id":"5834ecf7432d92675bde9d85"
}

Aquí estoy tratando de lograr la salida en el siguiente formato

Rendimiento esperado

[{
   "_id": "5834ecf7432d92675bde9d83",
   "name": "NIFT Hyderabad",
   "university_id": "5834ecf7432d92675bde9d82",
   "departments": [{
        "_id": "5834ecf7432d92675bde9d84",
        "department_name": "CSE",
        "college_id": "5834ecf7432d92675bde9d83",
        "sections": [{
            "_id": "5834ecf7432d92675bde9d86",
            "section_name": "A",
            "students": "56",
            "department_id": "5834ecf7432d92675bde9d84"
        }, {
            "_id": "5834ecf7432d92675bde9d87",
            "section_name": "B",
            "students": "60",
            "department_id": "5834ecf7432d92675bde9d84"
        }]
    },
    {
        "_id": "5834ecf7432d92675bde9d85",
        "department_name": "Mechanical",
        "college_id": "5834ecf7432d92675bde9d83",
        "sections": [{
                "_id": "5834ecf7432d92675bde9d86",
                "section_name": "A",
                "students": "55",
                "department_id": "5834ecf7432d92675bde9d85"
            },
            {
                "_id": "5834ecf7432d92675bde9d87",
                "section_name": "B",
                "students": "44",
                "department_id": "5834ecf7432d92675bde9d85"
            }
        ]
    }
  ]
}]

Pero, estoy obteniendo el departamento y las secciones en matrices separadas para la universidad, pero no puedo obtener el formato anterior

Consulta

db.college.aggregate([
            {"$match": { "university_id": "5834ecf7432d92675bde9d82" } },
            {"$lookup": {
            "localField": "_id",
            "from": "departments",
            "foreignField": "college_id",
            "as": "departments"
        }},
        {"$unwind":"$departments"},
        {$group : {_id : "$_id", departments : {$push : "$departments" }}},
        {"$lookup": {
        "localField": "departments._id",
        "from": "sections",
        "foreignField": "department_id",
        "as": "sections"}
        }
        ])

¿Alguien puede ayudarme a resolver este problema? Será muy útil para mí.