matplotlib
diff --git a/‎doc/api/next_api_changes/behavior/xxxxx-DS.rst
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diff --git a/‎lib/matplotlib/category.py
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diff --git a/‎lib/matplotlib/cm.py
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diff --git a/‎lib/matplotlib/tests/test_category.py
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@@ -14,3 +14,7 @@ convert data of more than one dimension (e.g. when plotting images the data is 2
 If a conversion interface raises an error when given ``None`` or 2D data as described
 above, this error will be re-raised when a user tries to use one of the newly supported
 plotting methods with unit-ful data.
+
+If you have a custom conversion interface you want to forbid using with image data, the
+`~.units.ConversionInterface` methods that accept a ``units`` parameter should raise
+a `matplotlib.units.ConversionError` when given ``units=None``.
@@ -23,6 +23,14 @@
 _log = logging.getLogger(__name__)
 
 
+def _check_axis(axis):
+    if axis is None:
+        raise units.ConversionError(
+            "Categorical data does not support unit conversion that is not "
+            "attached to an x or y Axis."
+        )
+
+
 class StrCategoryConverter(units.ConversionInterface):
     @staticmethod
     def convert(value, unit, axis):
@@ -45,6 +53,7 @@ def convert(value, unit, axis):
         -------
         float or `~numpy.ndarray` of float
         """
+        _check_axis(axis)
         if unit is None:
             raise ValueError(
                 'Missing category information for StrCategoryConverter; '
@@ -77,6 +86,7 @@ def axisinfo(unit, axis):
             Information to support default tick labeling
 
         """
+        _check_axis(axis)
         StrCategoryConverter._validate_unit(unit)
         # locator and formatter take mapping dict because
         # args need to be pass by reference for updates
@@ -100,6 +110,7 @@ def default_units(data, axis):
         `.UnitData`
             object storing string to integer mapping
         """
+        _check_axis(axis)
         # the conversion call stack is default_units -> axis_info -> convert
         if axis.units is None:
             axis.set_units(UnitData(data))
 
@@ -529,6 +529,9 @@ def _strip_units(self, A):
         try:
             self._units = self._converter.default_units(A, None)
         except Exception as e:
+            if isinstance(e, munits.ConversionError):
+                raise e
+
             raise RuntimeError(
                 f'{self._converter} failed when trying to return the default units for '
                 'this image. This may be because support has not been '
@@ -538,6 +541,9 @@ def _strip_units(self, A):
         try:
             A = self._converter.convert(A, self._units, None)
         except Exception as e:
+            if isinstance(e, munits.ConversionError):
+                raise e
+
             raise munits.ConversionError(
                 f'{self._converter} failed when trying to convert the units for this '
                 'image. This may be because support has not been implemented '
 
@@ -9,6 +9,7 @@
 import matplotlib.pyplot as plt
 import matplotlib.category as cat
 from matplotlib.testing.decorators import check_figures_equal
+import matplotlib.units as munits
 
 
 class TestUnitData:
@@ -321,3 +322,13 @@ def test_set_lim():
     ax.plot(["a", "b", "c", "d"], [1, 2, 3, 4])
     with warnings.catch_warnings():
         ax.set_xlim("b", "c")
+
+
+def test_mappable_error():
+    data = data = [["one", "two"], ["three", "four"]]
+
+    fig, ax = plt.subplots()
+    with pytest.raises(munits.ConversionError,
+                       match=('Categorical data does not support unit conversion '
+                              'that is not attached to an x or y Axis.')):
+        ax.imshow(data)