Add C document to sphinx, fix arrow. (#8300)

- Group C API.
- Add C API sphinx doc.
- Consistent use of `OptionalArg` and the parameter name `config`.
- Remove call to deprecated functions in demo.
- Fix some formatting errors.
- Add links to c examples in the document (only visible with doxygen pages)
- Fix arrow.

doc/c.rst

@@ -6,7 +6,59 @@ XGBoost implements a set of C API designed for various bindings, we maintain its
 and the CMake/make build interface.  See :doc:`/tutorials/c_api_tutorial` for an
 introduction and ``demo/c-api/`` for related examples.  Also one can generate doxygen
 document by providing ``-DBUILD_C_DOC=ON`` as parameter to ``CMake`` during build, or
-simply look at function comments in ``include/xgboost/c_api.h``.
+simply look at function comments in ``include/xgboost/c_api.h``. The reference is exported
+to sphinx with the help of breathe, which doesn't contain links to examples but might be
+easier to read. For the original doxygen pages please visit:
 
 * `C API documentation (latest master branch) <https://xgboost.readthedocs.io/en/latest/dev/c__api_8h.html>`_
 * `C API documentation (last stable release) <https://xgboost.readthedocs.io/en/stable/dev/c__api_8h.html>`_
+
+***************
+C API Reference
+***************
+
+.. contents::
+  :backlinks: none
+  :local:
+
+Library
+=======
+
+.. doxygengroup:: Library
+   :project: xgboost
+
+DMatrix
+=======
+
+.. doxygengroup:: DMatrix
+   :project: xgboost
+
+Streaming
+---------
+
+.. doxygengroup:: Streaming
+   :project: xgboost
+
+Booster
+=======
+
+.. doxygengroup:: Booster
+   :project: xgboost
+
+Prediction
+----------
+
+.. doxygengroup:: Prediction
+   :project: xgboost
+
+Serialization
+-------------
+
+.. doxygengroup:: Serialization
+   :project: xgboost
+
+Collective
+==========
+
+.. doxygengroup:: Collective
+   :project: xgboost