diff --git a/polars/polars-core/src/schema.rs b/polars/polars-core/src/schema.rs
index aae30a200273..aed70212943a 100644
--- a/polars/polars-core/src/schema.rs
+++ b/polars/polars-core/src/schema.rs
@@ -131,6 +131,10 @@ impl Schema {
             .ok_or_else(|| PolarsError::NotFound(name.to_string().into()))
     }
 
+    pub fn remove(&mut self, name: &str) -> Option<DataType> {
+        self.inner.remove(name)
+    }
+
     pub fn get_full(&self, name: &str) -> Option<(usize, &String, &DataType)> {
         self.inner.get_full(name)
     }
@@ -170,8 +174,18 @@ impl Schema {
         Some(())
     }
 
-    pub fn with_column(&mut self, name: String, dtype: DataType) {
-        self.inner.insert(name, dtype);
+    /// Insert a new column in the [`Schema`]
+    ///
+    /// If an equivalent name already exists in the schema: the name remains and
+    /// retains in its place in the order, its corresponding value is updated
+    /// with [`DataType`] and the older dtype is returned inside `Some(_)`.
+    ///
+    /// If no equivalent key existed in the map: the new name-dtype pair is
+    /// inserted, last in order, and `None` is returned.
+    ///
+    /// Computes in **O(1)** time (amortized average).
+    pub fn with_column(&mut self, name: String, dtype: DataType) -> Option<DataType> {
+        self.inner.insert(name, dtype)
     }
 
     pub fn merge(&mut self, other: Self) {
diff --git a/polars/polars-lazy/polars-plan/src/logical_plan/builder.rs b/polars/polars-lazy/polars-plan/src/logical_plan/builder.rs
index f19e4a65c391..1c5cbf5a0669 100644
--- a/polars/polars-lazy/polars-plan/src/logical_plan/builder.rs
+++ b/polars/polars-lazy/polars-plan/src/logical_plan/builder.rs
@@ -378,7 +378,7 @@ impl LogicalPlanBuilder {
 
             for fld in other_schema.iter_fields() {
                 if schema.get(fld.name()).is_none() {
-                    schema.with_column(fld.name, fld.dtype)
+                    schema.with_column(fld.name, fld.dtype);
                 }
             }
         }
@@ -550,7 +550,7 @@ impl LogicalPlanBuilder {
                 if let Expr::Column(name) = e {
                     if let Some(DataType::List(inner)) = schema.get(name) {
                         let inner = *inner.clone();
-                        schema.with_column(name.to_string(), inner)
+                        schema.with_column(name.to_string(), inner);
                     }
 
                     (**name).to_owned()
diff --git a/polars/polars-lazy/polars-plan/src/logical_plan/functions/mod.rs b/polars/polars-lazy/polars-plan/src/logical_plan/functions/mod.rs
index 1491ae98c803..5024b722692f 100644
--- a/polars/polars-lazy/polars-plan/src/logical_plan/functions/mod.rs
+++ b/polars/polars-lazy/polars-plan/src/logical_plan/functions/mod.rs
@@ -107,7 +107,7 @@ impl FunctionNode {
                             if let DataType::Struct(flds) = dtype {
                                 for fld in flds {
                                     new_schema
-                                        .with_column(fld.name().clone(), fld.data_type().clone())
+                                        .with_column(fld.name().clone(), fld.data_type().clone());
                                 }
                             } else {
                                 return Err(PolarsError::ComputeError(
@@ -115,7 +115,7 @@ impl FunctionNode {
                                 ));
                             }
                         } else {
-                            new_schema.with_column(name.clone(), dtype.clone())
+                            new_schema.with_column(name.clone(), dtype.clone());
                         }
                     }
 
diff --git a/polars/polars-lazy/polars-plan/src/logical_plan/optimizer/projection_pushdown/mod.rs b/polars/polars-lazy/polars-plan/src/logical_plan/optimizer/projection_pushdown/mod.rs
index df0217cd14fa..f707b7934f03 100644
--- a/polars/polars-lazy/polars-plan/src/logical_plan/optimizer/projection_pushdown/mod.rs
+++ b/polars/polars-lazy/polars-plan/src/logical_plan/optimizer/projection_pushdown/mod.rs
@@ -134,7 +134,7 @@ fn update_scan_schema(
         new_cols.sort_unstable_by_key(|item| item.0);
     }
     for item in new_cols {
-        new_schema.with_column(item.1.clone(), item.2.clone())
+        new_schema.with_column(item.1.clone(), item.2.clone());
     }
     Ok(new_schema)
 }
@@ -718,7 +718,7 @@ impl ProjectionPushDown {
                     let other_schema = lp_arena.get(*node).schema(lp_arena);
                     for fld in other_schema.iter_fields() {
                         if new_schema.get(fld.name()).is_none() {
-                            new_schema.with_column(fld.name, fld.dtype)
+                            new_schema.with_column(fld.name, fld.dtype);
                         }
                     }
                 }
diff --git a/polars/polars-lazy/polars-plan/src/logical_plan/schema.rs b/polars/polars-lazy/polars-plan/src/logical_plan/schema.rs
index 9598921ded95..d1b83b7e8c9a 100644
--- a/polars/polars-lazy/polars-plan/src/logical_plan/schema.rs
+++ b/polars/polars-lazy/polars-plan/src/logical_plan/schema.rs
@@ -197,7 +197,7 @@ pub(crate) fn det_join_schema(
 
             for (name, dtype) in schema_left.iter() {
                 names.insert(name.as_str());
-                new_schema.with_column(name.to_string(), dtype.clone())
+                new_schema.with_column(name.to_string(), dtype.clone());
             }
 
             // make sure that expression are assigned to the schema
diff --git a/polars/polars-lazy/src/frame/csv.rs b/polars/polars-lazy/src/frame/csv.rs
index 52fc9a6f4bad..5788610b41b2 100644
--- a/polars/polars-lazy/src/frame/csv.rs
+++ b/polars/polars-lazy/src/frame/csv.rs
@@ -255,7 +255,7 @@ impl<'a> LazyCsvReader<'a> {
         // the dtypes set may be for the new names, so update again
         if let Some(overwrite_schema) = self.schema_overwrite {
             for (name, dtype) in overwrite_schema.iter() {
-                schema.with_column(name.clone(), dtype.clone())
+                schema.with_column(name.clone(), dtype.clone());
             }
         }
 
diff --git a/polars/polars-lazy/src/frame/mod.rs b/polars/polars-lazy/src/frame/mod.rs
index 22190c170716..ba3299ce8011 100644
--- a/polars/polars-lazy/src/frame/mod.rs
+++ b/polars/polars-lazy/src/frame/mod.rs
@@ -278,7 +278,9 @@ impl LazyFrame {
             // schema after renaming
             for (old, new) in existing2.iter().zip(new2.iter()) {
                 let dtype = old_schema.try_get(old)?;
-                new_schema.with_column(new.clone(), dtype.clone());
+                if new_schema.with_column(new.clone(), dtype.clone()).is_none() {
+                    new_schema.remove(old);
+                }
             }
             Ok(Arc::new(new_schema))
         };
@@ -297,7 +299,12 @@ impl LazyFrame {
                 let columns = std::mem::take(df.get_columns_mut());
                 DataFrame::new(columns)
             },
-            None,
+            // Don't allow optimizations. Swapping names are opaque to the optimizer
+            AllowedOptimizations {
+                projection_pushdown: false,
+                predicate_pushdown: false,
+                ..Default::default()
+            },
             Some(Arc::new(udf_schema)),
             Some("RENAME_SWAPPING"),
         )
@@ -343,7 +350,7 @@ impl LazyFrame {
                 cols.truncate(cols.len() - (existing.len() - removed_count));
                 Ok(df)
             },
-            None,
+            Default::default(),
             Some(Arc::new(udf_schema)),
             Some("RENAME"),
         )
@@ -1118,7 +1125,7 @@ impl LazyFrame {
     pub fn map<F>(
         self,
         function: F,
-        optimizations: Option<AllowedOptimizations>,
+        optimizations: AllowedOptimizations,
         schema: Option<Arc<dyn UdfSchema>>,
         name: Option<&'static str>,
     ) -> LazyFrame
@@ -1130,7 +1137,7 @@ impl LazyFrame {
             .get_plan_builder()
             .map(
                 function,
-                optimizations.unwrap_or_default(),
+                optimizations,
                 schema,
                 name.unwrap_or("ANONYMOUS UDF"),
             )
@@ -1206,7 +1213,7 @@ impl LazyFrame {
                     Ok(df)
                 }
             },
-            Some(opt),
+            opt,
             Some(Arc::new(udf_schema)),
             Some("WITH ROW COUNT"),
         )
diff --git a/polars/polars-lazy/src/tests/predicate_queries.rs b/polars/polars-lazy/src/tests/predicate_queries.rs
index 6f4450f07421..5f774d030aab 100644
--- a/polars/polars-lazy/src/tests/predicate_queries.rs
+++ b/polars/polars-lazy/src/tests/predicate_queries.rs
@@ -107,7 +107,7 @@ fn filter_blocked_by_map() -> PolarsResult<()> {
     };
     let q = df
         .lazy()
-        .map(|df| Ok(df), Some(allowed), None, None)
+        .map(|df| Ok(df), allowed, None, None)
         .filter(col("A").gt(lit(2i32)));
 
     assert!(!predicate_at_scan(q.clone()));
diff --git a/py-polars/polars/internals/lazyframe/frame.py b/py-polars/polars/internals/lazyframe/frame.py
index 62bd75c508a9..ae1898dac461 100644
--- a/py-polars/polars/internals/lazyframe/frame.py
+++ b/py-polars/polars/internals/lazyframe/frame.py
@@ -2736,6 +2736,11 @@ def rename(self: LDF, mapping: dict[str, str]) -> LDF:
         mapping
             Key value pairs that map from old name to new name.
 
+        Notes
+        -----
+        If names are swapped. E.g. 'A' points to 'B' and 'B' points to 'A', polars
+        will block projection and predicate pushdowns at this node.
+
         Examples
         --------
         >>> df = pl.DataFrame(
diff --git a/py-polars/src/dataframe.rs b/py-polars/src/dataframe.rs
index 1eecb55f1932..27f8acaffe43 100644
--- a/py-polars/src/dataframe.rs
+++ b/py-polars/src/dataframe.rs
@@ -77,7 +77,7 @@ impl PyDataFrame {
                         *dtype_ = dtype;
                     }
                 } else {
-                    schema.with_column(name, dtype)
+                    schema.with_column(name, dtype);
                 }
             }
         }
diff --git a/py-polars/src/lazy/dataframe.rs b/py-polars/src/lazy/dataframe.rs
index 2cc99e71129a..3062cea50935 100644
--- a/py-polars/src/lazy/dataframe.rs
+++ b/py-polars/src/lazy/dataframe.rs
@@ -879,7 +879,7 @@ impl PyLazyFrame {
 
         let udf_schema =
             schema.map(move |s| Arc::new(move |_: &Schema| Ok(s.clone())) as Arc<dyn UdfSchema>);
-        ldf.map(function, Some(opt), udf_schema, None).into()
+        ldf.map(function, opt, udf_schema, None).into()
     }
 
     pub fn drop_columns(&self, cols: Vec<String>) -> Self {
diff --git a/py-polars/tests/unit/test_df.py b/py-polars/tests/unit/test_df.py
index 15eeed8d02b5..77ce81090734 100644
--- a/py-polars/tests/unit/test_df.py
+++ b/py-polars/tests/unit/test_df.py
@@ -1915,6 +1915,41 @@ def test_rename_swap() -> None:
     )
     assert out.frame_equal(expected)
 
+    # 6195
+    ldf = pl.DataFrame(
+        {
+            "weekday": [
+                1,
+            ],
+            "priority": [
+                2,
+            ],
+            "roundNumber": [
+                3,
+            ],
+            "flag": [
+                4,
+            ],
+        }
+    ).lazy()
+
+    # Rename some columns (note: swapping two columns)
+    rename_dict = {
+        "weekday": "priority",
+        "priority": "weekday",
+        "roundNumber": "round_number",
+    }
+    ldf = ldf.rename(rename_dict)
+
+    # Select some columns
+    ldf = ldf.select(["priority", "weekday", "round_number"])
+
+    assert ldf.collect().to_dict(False) == {
+        "priority": [1],
+        "weekday": [2],
+        "round_number": [3],
+    }
+
 
 def test_rename_same_name() -> None:
     df = pl.DataFrame(