Although it doesn't start with _ and is definitely necessary as codecs
call it.

AFAIR, this code was added by Martin as optimization.

The function is a direct interface to PyUnicode_BuildEncodingMap() which
builds either a dictionary or EncodingMap object for use by the charmap
codec.

It is primarily used by the gencodec.py script.

There are numerous undocumented implementation objects that are either private and probably should start with _ but do not, or are semi-private and perhaps left public for the expert user willing to read code. But, being undocumented, it is hard to tell. codecs.charmap_build strikes me as rather private, hence should not be documented. So I would be inclined to close this.

I see charmap_build used in encodings too, so it may be useful to make it officially public for the use of third-party encodings.

msg120987 states the opposite to msg112747 so what do we do with this issue?

I've found it rather ugly to even understand from the lack of code comments for the C level API it's wrapping. If nothing else, comments explaining the usage and purpose might be helpful.

I do not see much purpose in documenting and testing this one function alone.  I think that all functions in _codecsmodule.c should be looked at to see if they are private or public, but I'm certainly not volunteering to do it.