PR Summary

Start addresses the following section of #16181:

Move all the internal cbook functions to (a/multiple) private module(s).

from matplotlib import _api

@_api.deprecated("3.2")
def compare_versions(a, b):

reads better than the current

from matplotlib import cbook

@cbook._deprecated("3.2")
def compare_versions(a, b):

and it does not pollute the namespace with the public package name.

This PR does:

• Move cbook._check_in_list() to _api.check_in_list()