BUG: .str.startswith(..., na=False) consistency between categorical and string series (again)

[dlimpid]

Code Sample, a copy-pastable example

>>> import pandas as pd
>>> import numpy as np
>>> s = pd.Series(["a", "b", np.nan], index=["a", "b", np.nan], dtype="category")
>>> df = pd.DataFrame().assign(
...     cat_contains=s.str.contains("a", na=False),
...     cat_startswith=s.str.startswith("a", na=False),
...     cat_endswith=s.str.endswith("a", na=False),
...     str_contains=s.astype("string").str.contains("a", na=False),
...     str_startswith=s.astype("string").str.startswith("a", na=False),
...     str_endswith=s.astype("string").str.endswith("a", na=False),
... )
>>> df.info()
<class 'pandas.core.frame.DataFrame'>
Index: 3 entries, a to nan
Data columns (total 6 columns):
 #   Column          Non-Null Count  Dtype
---  ------          --------------  -----
 0   cat_contains    3 non-null      bool
 1   cat_startswith  2 non-null      object
 2   cat_endswith    2 non-null      object
 3   str_contains    3 non-null      boolean
 4   str_startswith  3 non-null      boolean
 5   str_endswith    3 non-null      boolean
dtypes: bool(1), boolean(3), object(2)
memory usage: 93.0+ bytes
>>> df
     cat_contains cat_startswith cat_endswith  str_contains  str_startswith  str_endswith
a            True           True         True          True            True          True
b           False          False        False         False           False         False
NaN         False            NaN          NaN         False           False         False

Problem description

.str.startswith(..., na=False) and .str.endswith should make missing values False when the calling series is of type categorical just like it does for string series.

Similar to #22158, but .str.contains works here.

Expected Output

>>> df
     cat_contains cat_startswith cat_endswith  str_contains  str_startswith  str_endswith
a            True           True         True          True            True          True
b           False          False        False         False           False         False
NaN         False          False        False         False           False         False

Output of pd.show_versions()

Using conda env with conda create -n pandas112 -c conda-forge pandas=1.1.2

INSTALLED VERSIONS

commit : 2a7d332
python : 3.8.5.final.0
python-bits : 64
OS : Windows
OS-release : 10
Version : 10.0.19041
machine : AMD64
processor : Intel64 Family 6 Model 142 Stepping 10, GenuineIntel
byteorder : little
LC_ALL : None
LANG : None
LOCALE : Korean_Korea.949

pandas : 1.1.2
numpy : 1.19.1
pytz : 2020.1
dateutil : 2.8.1
pip : 20.2.3
setuptools : 49.6.0.post20200814
Cython : None
pytest : None
hypothesis : None
sphinx : None
blosc : None
feather : None
xlsxwriter : None
lxml.etree : None
html5lib : None
pymysql : None
psycopg2 : None
jinja2 : None
IPython : None
pandas_datareader: None
bs4 : None
bottleneck : None
fsspec : None
fastparquet : None
gcsfs : None
matplotlib : None
numexpr : None
odfpy : None
openpyxl : None
pandas_gbq : None
pyarrow : None
pytables : None
pyxlsb : None
s3fs : None
scipy : None
sqlalchemy : None
tables : None
tabulate : None
xarray : None
xlrd : None
xlwt : None
numba : None

[asishm]

did some digging into this:

In [15]: pd.Series(["a", "b", 1, np.nan,1, pd.NA]).str.startswith('a')
Out[15]: 
0     True
1    False
2      NaN
3      NaN
4      NaN
5      NaN
dtype: object

In [16]: pd.Series(["a", "b", 1, np.nan,1, pd.NA]).str.startswith('a', na=True)
Out[16]: 
0     True
1    False
2     True
3     True
4     True
5     True
dtype: bool

In [17]: pd.Series(["a", "b", 1, np.nan,1, pd.NA], dtype='category').str.startswith('a')
Out[17]: 
0     True
1    False
2      NaN
3      NaN
4      NaN
5      NaN
dtype: object

In [18]: pd.Series(["a", "b", 1, np.nan,1, pd.NA], dtype='category').str.startswith('a', na=True)
Out[18]: 
0     True
1    False
2     True
3      NaN
4     True
5      NaN
dtype: object

It seems like na=True is not respected only if it is a Categorical dtype and for entries that are NA (np.nan, pd.NA, None)