diff --git a/sphinx/domains/cpp.py b/sphinx/domains/cpp.py
index 20c0f0074..9e8624df8 100644
--- a/sphinx/domains/cpp.py
+++ b/sphinx/domains/cpp.py
@@ -5469,12 +5469,12 @@ class CPPDomain(Domain):
     name = 'cpp'
     label = 'C++'
     object_types = {
-        'class':      ObjType(l_('class'),      'class',             'type', 'typeOrConcept'),
-        'function':   ObjType(l_('function'),   'function',  'func', 'type', 'typeOrConcept'),
+        'class':      ObjType(l_('class'),      'class',             'type', 'identifier'),
+        'function':   ObjType(l_('function'),   'function',  'func', 'type', 'identifier'),
         'member':     ObjType(l_('member'),     'member',    'var'),
-        'type':       ObjType(l_('type'),                            'type', 'typeOrConcept'),
-        'concept':    ObjType(l_('concept'),    'concept',                   'typeOrConcept'),
-        'enum':       ObjType(l_('enum'),       'enum',              'type', 'typeOrConcept'),
+        'type':       ObjType(l_('type'),                            'type', 'identifier'),
+        'concept':    ObjType(l_('concept'),    'concept',                   'identifier'),
+        'enum':       ObjType(l_('enum'),       'enum',              'type', 'identifier'),
         'enumerator': ObjType(l_('enumerator'), 'enumerator')
     }
 
diff --git a/tests/test_ext_intersphinx.py b/tests/test_ext_intersphinx.py
index 76f4cef0a..4965108ef 100644
--- a/tests/test_ext_intersphinx.py
+++ b/tests/test_ext_intersphinx.py
@@ -232,12 +232,6 @@ def test_missing_reference_cppdomain(tempdir, app, status, warning):
             ' href="https://docs.python.org/index.html#cpp_foo_bar"'
             ' title="(in foo v2.0)"><code class="xref cpp cpp-class docutils literal">'
             '<span class="pre">Bar</span></code></a>' in html)
-    # assert ('<a class="reference external"'
-    #         ' href="https://docs.python.org/index.html#std"'
-    #         ' title="(in foo v2.0)">std</a>' in html)
-    # assert ('<a class="reference external"'
-    #         ' href="https://docs.python.org/index.html#std_uint8_t"'
-    #         ' title="(in foo v2.0)">uint8_t</a>' in html)
 
 
 def test_missing_reference_jsdomain(tempdir, app, status, warning):