code: def _add_approx_dataframe_index_col(
      es, target_dataframe_name, cutoffs, path
  ):
      """
      Add a column to the cutoff df linking it to the dataframe at the end of the
      path.
  
      Return the updated cutoff df and the name of this column. The name will
      consist of the columns which were joined through.
      """
      last_child_col = "instance_id"
      last_parent_col = es[target_dataframe_name].ww.index
  
      for _, relationship in path:
          child_cols = [
              last_parent_col,
              relationship._child_column_name,
          ]
          child_df = es[relationship.child_name][child_cols]
  
          # Rename relationship.child_column to include the columns we have
          # joined through.
          new_col_name = "%s.%s" % (
              last_child_col,
              relationship._child_column_name,
          )
          to_rename = {relationship._child_column_name: new_col_name}
          child_df = child_df.rename(columns=to_rename)
          cutoffs = cutoffs.merge(
              child_df,
              left_on=last_child_col,
              right_on=last_parent_col,
          )
  
          # These will be used in the next iteration.
          last_child_col = new_col_name
          last_parent_col = relationship._parent_column_name
  
      return cutoffs, new_col_name
  
imports: import logging
  import math
  import os
  import shutil
  import time
  import warnings
  from datetime import datetime
  import cloudpickle
  import numpy
  import pandas
  from woodwork.logical_types import Age, AgeNullable, Boolean, BooleanNullable, Integer, IntegerNullable
  from featuretools.computational_backends.feature_set import FeatureSet
  from featuretools.computational_backends.feature_set_calculator import FeatureSetCalculator
  from featuretools.computational_backends.utils import _check_cutoff_time_type, _validate_cutoff_time, bin_cutoff_times, create_client_and_cluster, gather_approximate_features, gen_empty_approx_features_df, get_ww_types_from_features, save_csv_decorator
  from featuretools.entityset.relationship import RelationshipPath
  from featuretools.feature_base import AggregationFeature, FeatureBase
  from featuretools.utils import Trie
  from featuretools.utils.gen_utils import Library, import_or_none, import_or_raise, is_instance, make_tqdm_iterator
label_desc: Use image node_id 6 for calling a global function with example usage: _add_approx_dataframe_index_col(es, target_dataframe_name, cutoffs, path) and returns: cutoffs,  new_col_name