diff --git a/src/pydiverse/transform/_internal/pipe/table.py b/src/pydiverse/transform/_internal/pipe/table.py
index 03e0855..0927a4d 100644
--- a/src/pydiverse/transform/_internal/pipe/table.py
+++ b/src/pydiverse/transform/_internal/pipe/table.py
@@ -194,7 +194,7 @@ def update(self, vb: Verb, rcache: Cache | None = None):
             self._update(new_select=vb.select)
 
         elif isinstance(vb, Rename):
-            self.update(
+            self._update(
                 new_cols={
                     (new_name if (new_name := vb.name_map.get(name)) else name): col
                     for name, col in self.cols.items()
diff --git a/src/pydiverse/transform/_internal/pipe/verbs.py b/src/pydiverse/transform/_internal/pipe/verbs.py
index 25b15cc..2aa4853 100644
--- a/src/pydiverse/transform/_internal/pipe/verbs.py
+++ b/src/pydiverse/transform/_internal/pipe/verbs.py
@@ -276,11 +276,11 @@ def rename(table: Table, name_map: dict[str, str]) -> Pipeable:
 
     if d := set(name_map) - set(table._cache.cols):
         raise ValueError(
-            f"no column with name `{next(d)}` in table `{table._ast.name}`"
+            f"no column with name `{next(iter(d))}` in table `{table._ast.name}`"
         )
 
-    if d := (set(table._cache.cols) - set(name_map)) | set(name_map.values()):
-        raise ValueError(f"duplicate column name `{next(d)}`")
+    if d := (set(table._cache.cols) - set(name_map)) & set(name_map.values()):
+        raise ValueError(f"duplicate column name `{next(iter(d))}`")
 
     new._cache.update(new._ast)
     return new
@@ -292,6 +292,8 @@ def mutate(**kwargs: ColExpr) -> Pipeable: ...
 
 @verb
 def mutate(table: Table, **kwargs: ColExpr) -> Pipeable:
+    if len(kwargs) == 0:
+        return table
     return table >> _mutate(*map(list, zip(*kwargs.items(), strict=True)))
 
 
@@ -302,9 +304,6 @@ def _mutate(
     values: list[ColExpr],
     uuids: list[uuid.UUID] | None = None,
 ) -> Pipeable:
-    if len(names) == 0:
-        return table
-
     new = copy.copy(table)
     if uuids is None:
         uuids = [uuid.uuid1() for _ in names]
diff --git a/tests/test_core.py b/tests/test_core.py
index 367a487..f7b4d52 100644
--- a/tests/test_core.py
+++ b/tests/test_core.py
@@ -5,7 +5,7 @@
 
 from pydiverse.transform import C, Table
 from pydiverse.transform._internal.pipe.pipeable import Pipeable, inverse_partial, verb
-from pydiverse.transform._internal.pipe.verbs import join, mutate, select
+from pydiverse.transform._internal.pipe.verbs import mutate, select
 
 
 @pytest.fixture
@@ -27,11 +27,8 @@ def test_getattr(self, tbl1):
             _ = tbl1.colXXX
 
     def test_getitem(self, tbl1):
-        assert tbl1.col1 is tbl1["col1"]
-        assert tbl1.col2 is tbl1["col2"]
-
-        assert tbl1.col2 is tbl1[tbl1.col2.name]
-        assert tbl1.col2 is tbl1[C.col2.name]
+        assert tbl1.col1._uuid == tbl1["col1"]._uuid
+        assert tbl1.col2._uuid is tbl1["col2"]._uuid
 
     def test_iter(self, tbl1, tbl2):
         assert repr(list(tbl1)) == repr([tbl1.col1, tbl1.col2])
@@ -39,11 +36,6 @@ def test_iter(self, tbl1, tbl2):
 
         assert repr(list(tbl2 >> select(tbl2.col2))) == repr([tbl2.col2])
 
-        joined = tbl1 >> join(
-            tbl2 >> select(tbl2.col3), tbl1.col1 == tbl2.col2, "left", suffix="_2"
-        )
-        assert repr(list(joined)) == repr([tbl1.col1, tbl1.col2, joined.col3_2])
-
     def test_dir(self, tbl1):
         assert dir(tbl1) == ["col1", "col2"]
         assert dir(tbl1 >> mutate(x=tbl1.col1)) == ["col1", "col2", "x"]