C functions are not shown in sidebar

[cc00010]

Describe the bug

.. py:function:: or .. function:: shows function names in sidebar. But .. c:function:: does not.

How to Reproduce

.. function:: works_as_expected()

.. c:function:: void no_symbol()

Environment Information

Platform:              darwin; (macOS-13.3-arm64-arm-64bit)
Python version:        3.9.7 (default, Sep  3 2021, 04:31:11) 
[Clang 12.0.5 (clang-1205.0.22.9)])
Python implementation: CPython
Sphinx version:        6.2.1
Docutils version:      0.18.1
Jinja2 version:        3.1.2
Pygments version:      2.13.0

Sphinx extensions

No response

Additional context

No response

[cc00010]

Same issue with cpp. But javascript works as intended.

[AA-Turner]

Happy to accept a PR. The C domain CObject would need to implement _toc_entry_name and _object_hierarchy_parts.

A