In MongoDB, what is the correct way to update a referenced object upon saving another object?

I am using Mongoose ODM. Having 2 schemas - I would like to ensure that when I create a new Supplier, that it’s ID is added to the reference object array - suppliers on SupplierType.

Supplier:

name: {
    type: String,
    required: true,
    minlength: 2,
    maxlength: 255,
  },
  ...
supplier_type: { type: ObjectId, ref: 'SupplierType' },
}

SupplierType:

name: {
    type: String,
    required: true,
    minlength: 2,
    maxlength: 50,
  },
suppliers: [{ type: ObjectId, ref: 'Supplier' }],
};

When a user makes a POST to create a new Supplier - I execute the following:

const supplier = new Supplier({
    name: req.body.name,
    ...
    supplier_type: req.body.supplier_type,
  });
await supplier.save();

This, of course, does not update/add the new Supplier to the suppliers array on Supplier Type, so I do the following:

// Save the supplier
await supplier.save();

// Add the new Supplier's ID to the `suppliers` array on the `Supplier Type`
await SupplierType.update(
    {
      _id: supplier.supplier_type,
    },
    { $push: { suppliers: supplier.id } }
);

I’d like to know if this is best practice and if not, I’d like to know the best way to achieve the same goal?