diff --git a/narwhals/_arrow/expr.py b/narwhals/_arrow/expr.py
index c80238ec4f..b97b50ac0f 100644
--- a/narwhals/_arrow/expr.py
+++ b/narwhals/_arrow/expr.py
@@ -2,7 +2,6 @@
 
 from typing import TYPE_CHECKING
 from typing import Any
-from typing import Callable
 from typing import Literal
 from typing import Sequence
 
@@ -22,6 +21,9 @@
 
     from narwhals._arrow.dataframe import ArrowDataFrame
     from narwhals._arrow.namespace import ArrowNamespace
+    from narwhals._compliant.typing import AliasNames
+    from narwhals._compliant.typing import EvalNames
+    from narwhals._compliant.typing import EvalSeries
     from narwhals._expression_parsing import ExprMetadata
     from narwhals.utils import Version
     from narwhals.utils import _FullContext
@@ -32,12 +34,12 @@ class ArrowExpr(EagerExpr["ArrowDataFrame", ArrowSeries]):
 
     def __init__(
         self: Self,
-        call: Callable[[ArrowDataFrame], Sequence[ArrowSeries]],
+        call: EvalSeries[ArrowDataFrame, ArrowSeries],
         *,
         depth: int,
         function_name: str,
-        evaluate_output_names: Callable[[ArrowDataFrame], Sequence[str]],
-        alias_output_names: Callable[[Sequence[str]], Sequence[str]] | None,
+        evaluate_output_names: EvalNames[ArrowDataFrame],
+        alias_output_names: AliasNames | None,
         backend_version: tuple[int, ...],
         version: Version,
         call_kwargs: dict[str, Any] | None = None,
@@ -57,7 +59,7 @@ def __init__(
     @classmethod
     def from_column_names(
         cls: type[Self],
-        evaluate_column_names: Callable[[ArrowDataFrame], Sequence[str]],
+        evaluate_column_names: EvalNames[ArrowDataFrame],
         /,
         *,
         context: _FullContext,
@@ -139,11 +141,7 @@ def cum_sum(self: Self, *, reverse: bool) -> Self:
     def shift(self: Self, n: int) -> Self:
         return self._reuse_series("shift", n=n)
 
-    def over(
-        self: Self,
-        partition_by: Sequence[str],
-        order_by: Sequence[str] | None,
-    ) -> Self:
+    def over(self, partition_by: Sequence[str], order_by: Sequence[str] | None) -> Self:
         assert self._metadata is not None  # noqa: S101
         if partition_by and not is_scalar_like(self._metadata.kind):
             msg = "Only aggregation or literal operations are supported in grouped `over` context for PyArrow."
diff --git a/narwhals/_compliant/expr.py b/narwhals/_compliant/expr.py
index 3e3f938410..b54ce6acd5 100644
--- a/narwhals/_compliant/expr.py
+++ b/narwhals/_compliant/expr.py
@@ -58,6 +58,9 @@
     from narwhals._compliant.namespace import CompliantNamespace
     from narwhals._compliant.namespace import EagerNamespace
     from narwhals._compliant.series import CompliantSeries
+    from narwhals._compliant.typing import AliasNames
+    from narwhals._compliant.typing import EvalNames
+    from narwhals._compliant.typing import EvalSeries
     from narwhals._expression_parsing import ExprKind
     from narwhals._expression_parsing import ExprMetadata
     from narwhals.dtypes import DType
@@ -84,21 +87,19 @@ class CompliantExpr(Protocol38[CompliantFrameT, CompliantSeriesOrNativeExprT_co]
     _implementation: Implementation
     _backend_version: tuple[int, ...]
     _version: Version
-    _evaluate_output_names: Callable[[CompliantFrameT], Sequence[str]]
-    _alias_output_names: Callable[[Sequence[str]], Sequence[str]] | None
+    _evaluate_output_names: EvalNames[CompliantFrameT]
+    _alias_output_names: AliasNames | None
     _metadata: ExprMetadata | None
 
     def __call__(
         self, df: CompliantFrameT
     ) -> Sequence[CompliantSeriesOrNativeExprT_co]: ...
     def __narwhals_expr__(self) -> None: ...
-    def __narwhals_namespace__(
-        self,
-    ) -> CompliantNamespace[CompliantFrameT, Self]: ...
+    def __narwhals_namespace__(self) -> CompliantNamespace[CompliantFrameT, Self]: ...
     @classmethod
     def from_column_names(
         cls,
-        evaluate_column_names: Callable[[CompliantFrameT], Sequence[str]],
+        evaluate_column_names: EvalNames[CompliantFrameT],
         /,
         *,
         context: _FullContext,
@@ -298,7 +299,7 @@ class DepthTrackingExpr(
     @classmethod
     def from_column_names(
         cls: type[Self],
-        evaluate_column_names: Callable[[CompliantFrameT], Sequence[str]],
+        evaluate_column_names: EvalNames[CompliantFrameT],
         /,
         *,
         context: _FullContext,
@@ -330,17 +331,17 @@ class EagerExpr(
     DepthTrackingExpr[EagerDataFrameT, EagerSeriesT],
     Protocol38[EagerDataFrameT, EagerSeriesT],
 ):
-    _call: Callable[[EagerDataFrameT], Sequence[EagerSeriesT]]
+    _call: EvalSeries[EagerDataFrameT, EagerSeriesT]
     _call_kwargs: dict[str, Any]
 
     def __init__(
         self: Self,
-        call: Callable[[EagerDataFrameT], Sequence[EagerSeriesT]],
+        call: EvalSeries[EagerDataFrameT, EagerSeriesT],
         *,
         depth: int,
         function_name: str,
-        evaluate_output_names: Callable[[EagerDataFrameT], Sequence[str]],
-        alias_output_names: Callable[[Sequence[str]], Sequence[str]] | None,
+        evaluate_output_names: EvalNames[EagerDataFrameT],
+        alias_output_names: AliasNames | None,
         implementation: Implementation,
         backend_version: tuple[int, ...],
         version: Version,
@@ -358,12 +359,12 @@ def __narwhals_expr__(self) -> None: ...
     @classmethod
     def _from_callable(
         cls,
-        func: Callable[[EagerDataFrameT], Sequence[EagerSeriesT]],
+        func: EvalSeries[EagerDataFrameT, EagerSeriesT],
         *,
         depth: int,
         function_name: str,
-        evaluate_output_names: Callable[[EagerDataFrameT], Sequence[str]],
-        alias_output_names: Callable[[Sequence[str]], Sequence[str]] | None,
+        evaluate_output_names: EvalNames[EagerDataFrameT],
+        alias_output_names: AliasNames | None,
         context: _FullContext,
         call_kwargs: dict[str, Any] | None = None,
     ) -> Self:
diff --git a/narwhals/_compliant/typing.py b/narwhals/_compliant/typing.py
index 41779df685..906d17570c 100644
--- a/narwhals/_compliant/typing.py
+++ b/narwhals/_compliant/typing.py
@@ -17,6 +17,7 @@
     from narwhals._compliant.expr import EagerExpr
     from narwhals._compliant.expr import LazyExpr
     from narwhals._compliant.expr import NativeExpr
+    from narwhals._compliant.namespace import CompliantNamespace
     from narwhals._compliant.namespace import EagerNamespace
     from narwhals._compliant.series import CompliantSeries
     from narwhals._compliant.series import EagerSeries
@@ -42,6 +43,7 @@
 CompliantDataFrameAny: TypeAlias = "CompliantDataFrame[Any, Any, Any]"
 CompliantLazyFrameAny: TypeAlias = "CompliantLazyFrame[Any, Any]"
 CompliantFrameAny: TypeAlias = "CompliantDataFrameAny | CompliantLazyFrameAny"
+CompliantNamespaceAny: TypeAlias = "CompliantNamespace[Any, Any]"
 
 DepthTrackingExprAny: TypeAlias = "DepthTrackingExpr[Any, Any]"
 
@@ -110,8 +112,18 @@
 LazyExprT_contra = TypeVar("LazyExprT_contra", bound=LazyExprAny, contravariant=True)
 
 AliasNames: TypeAlias = Callable[[Sequence[str]], Sequence[str]]
+"""A function aliasing a *sequence* of column names."""
+
 AliasName: TypeAlias = Callable[[str], str]
+"""A function aliasing a *single* column name."""
+
 EvalSeries: TypeAlias = Callable[
     [CompliantFrameT], Sequence[CompliantSeriesOrNativeExprT]
 ]
+"""A function from a `Frame` to a sequence of `Series`*.
+
+See [underwater unicorn magic](https://narwhals-dev.github.io/narwhals/how_it_works/).
+"""
+
 EvalNames: TypeAlias = Callable[[CompliantFrameT], Sequence[str]]
+"""A function from a `Frame` to a sequence of columns names *before* any aliasing takes place."""
diff --git a/narwhals/_compliant/when_then.py b/narwhals/_compliant/when_then.py
index 4a27cb77b5..696acfba41 100644
--- a/narwhals/_compliant/when_then.py
+++ b/narwhals/_compliant/when_then.py
@@ -24,6 +24,7 @@
     from typing_extensions import Self
     from typing_extensions import TypeAlias
 
+    from narwhals._compliant.typing import EvalSeries
     from narwhals.utils import Implementation
     from narwhals.utils import Version
     from narwhals.utils import _FullContext
@@ -82,7 +83,7 @@ def from_expr(cls, condition: ExprT, /, *, context: _FullContext) -> Self:
 
 
 class CompliantThen(CompliantExpr[FrameT, SeriesT], Protocol38[FrameT, SeriesT, ExprT]):
-    _call: Callable[[FrameT], Sequence[SeriesT]]
+    _call: EvalSeries[FrameT, SeriesT]
     _when_value: CompliantWhen[FrameT, SeriesT, ExprT]
     _function_name: str
     _depth: int
diff --git a/narwhals/_dask/expr.py b/narwhals/_dask/expr.py
index bfcadcf240..addc27f322 100644
--- a/narwhals/_dask/expr.py
+++ b/narwhals/_dask/expr.py
@@ -34,6 +34,8 @@
 
     from typing_extensions import Self
 
+    from narwhals._compliant.typing import EvalNames
+    from narwhals._compliant.typing import EvalSeries
     from narwhals._dask.dataframe import DaskLazyFrame
     from narwhals._dask.namespace import DaskNamespace
     from narwhals._expression_parsing import ExprMetadata
@@ -50,12 +52,12 @@ class DaskExpr(
 
     def __init__(
         self: Self,
-        call: Callable[[DaskLazyFrame], Sequence[dx.Series]],
+        call: EvalSeries[DaskLazyFrame, dx.Series],
         *,
         depth: int,
         function_name: str,
-        evaluate_output_names: Callable[[DaskLazyFrame], Sequence[str]],
-        alias_output_names: Callable[[Sequence[str]], Sequence[str]] | None,
+        evaluate_output_names: EvalNames[DaskLazyFrame],
+        alias_output_names: AliasNames | None,
         backend_version: tuple[int, ...],
         version: Version,
         # Kwargs with metadata which we may need in group-by agg
@@ -101,7 +103,7 @@ def func(df: DaskLazyFrame) -> list[dx.Series]:
     @classmethod
     def from_column_names(
         cls: type[Self],
-        evaluate_column_names: Callable[[DaskLazyFrame], Sequence[str]],
+        evaluate_column_names: EvalNames[DaskLazyFrame],
         /,
         *,
         context: _FullContext,
diff --git a/narwhals/_duckdb/expr.py b/narwhals/_duckdb/expr.py
index 5972b8f81b..dd54ed7bd9 100644
--- a/narwhals/_duckdb/expr.py
+++ b/narwhals/_duckdb/expr.py
@@ -34,6 +34,8 @@
     from typing_extensions import Self
 
     from narwhals._compliant.typing import AliasNames
+    from narwhals._compliant.typing import EvalNames
+    from narwhals._compliant.typing import EvalSeries
     from narwhals._duckdb.dataframe import DuckDBLazyFrame
     from narwhals._duckdb.namespace import DuckDBNamespace
     from narwhals._duckdb.typing import WindowFunction
@@ -51,10 +53,10 @@ class DuckDBExpr(LazyExpr["DuckDBLazyFrame", "duckdb.Expression"]):
 
     def __init__(
         self: Self,
-        call: Callable[[DuckDBLazyFrame], Sequence[duckdb.Expression]],
+        call: EvalSeries[DuckDBLazyFrame, duckdb.Expression],
         *,
-        evaluate_output_names: Callable[[DuckDBLazyFrame], Sequence[str]],
-        alias_output_names: Callable[[Sequence[str]], Sequence[str]] | None,
+        evaluate_output_names: EvalNames[DuckDBLazyFrame],
+        alias_output_names: AliasNames | None,
         backend_version: tuple[int, ...],
         version: Version,
     ) -> None:
@@ -168,7 +170,7 @@ def func(df: DuckDBLazyFrame) -> Sequence[duckdb.Expression]:
     @classmethod
     def from_column_names(
         cls: type[Self],
-        evaluate_column_names: Callable[[DuckDBLazyFrame], Sequence[str]],
+        evaluate_column_names: EvalNames[DuckDBLazyFrame],
         /,
         *,
         context: _FullContext,
@@ -243,10 +245,7 @@ def _with_alias_output_names(self, func: AliasNames | None, /) -> Self:
             version=self._version,
         )
 
-    def _with_window_function(
-        self: Self,
-        window_function: WindowFunction,
-    ) -> Self:
+    def _with_window_function(self, window_function: WindowFunction) -> Self:
         result = self.__class__(
             self._call,
             evaluate_output_names=self._evaluate_output_names,
diff --git a/narwhals/_expression_parsing.py b/narwhals/_expression_parsing.py
index d10f090728..f82ef55839 100644
--- a/narwhals/_expression_parsing.py
+++ b/narwhals/_expression_parsing.py
@@ -8,7 +8,6 @@
 from itertools import chain
 from typing import TYPE_CHECKING
 from typing import Any
-from typing import Callable
 from typing import Literal
 from typing import Sequence
 from typing import TypeVar
@@ -29,10 +28,13 @@
     from narwhals._compliant import CompliantExprT
     from narwhals._compliant import CompliantFrameT
     from narwhals._compliant import CompliantNamespace
+    from narwhals._compliant.typing import AliasNames
+    from narwhals._compliant.typing import CompliantExprAny
+    from narwhals._compliant.typing import CompliantFrameAny
+    from narwhals._compliant.typing import CompliantNamespaceAny
     from narwhals._compliant.typing import EagerNamespaceAny
+    from narwhals._compliant.typing import EvalNames
     from narwhals.expr import Expr
-    from narwhals.typing import CompliantDataFrame
-    from narwhals.typing import CompliantLazyFrame
     from narwhals.typing import IntoExpr
     from narwhals.typing import _1DArray
 
@@ -48,7 +50,7 @@ def is_expr(obj: Any) -> TypeIs[Expr]:
 
 def combine_evaluate_output_names(
     *exprs: CompliantExpr[CompliantFrameT, Any],
-) -> Callable[[CompliantFrameT], Sequence[str]]:
+) -> EvalNames[CompliantFrameT]:
     # Follow left-hand-rule for naming. E.g. `nw.sum_horizontal(expr1, expr2)` takes the
     # first name of `expr1`.
     if not is_compliant_expr(exprs[0]):  # pragma: no cover
@@ -61,9 +63,7 @@ def evaluate_output_names(df: CompliantFrameT) -> Sequence[str]:
     return evaluate_output_names
 
 
-def combine_alias_output_names(
-    *exprs: CompliantExpr[Any, Any],
-) -> Callable[[Sequence[str]], Sequence[str]] | None:
+def combine_alias_output_names(*exprs: CompliantExprAny) -> AliasNames | None:
     # Follow left-hand-rule for naming. E.g. `nw.sum_horizontal(expr1.alias(alias), expr2)` takes the
     # aliasing function of `expr1` and apply it to the first output name of `expr1`.
     if exprs[0]._alias_output_names is None:
@@ -91,9 +91,7 @@ def extract_compliant(
 
 
 def evaluate_output_names_and_aliases(
-    expr: CompliantExpr[Any, Any],
-    df: CompliantDataFrame[Any, Any, Any] | CompliantLazyFrame[Any, Any],
-    exclude: Sequence[str],
+    expr: CompliantExprAny, df: CompliantFrameAny, exclude: Sequence[str]
 ) -> tuple[Sequence[str], Sequence[str]]:
     output_names = expr._evaluate_output_names(df)
     if not output_names:
@@ -410,11 +408,8 @@ def infer_kind(obj: IntoExpr | _1DArray | object, *, str_as_lit: bool) -> ExprKi
 
 
 def apply_n_ary_operation(
-    plx: CompliantNamespace[Any, Any],
-    function: Any,
-    *comparands: IntoExpr,
-    str_as_lit: bool,
-) -> CompliantExpr[Any, Any]:
+    plx: CompliantNamespaceAny, function: Any, *comparands: IntoExpr, str_as_lit: bool
+) -> CompliantExprAny:
     compliant_exprs = (
         extract_compliant(plx, comparand, str_as_lit=str_as_lit)
         for comparand in comparands
diff --git a/narwhals/_pandas_like/expr.py b/narwhals/_pandas_like/expr.py
index 3f04123450..a2664598d2 100644
--- a/narwhals/_pandas_like/expr.py
+++ b/narwhals/_pandas_like/expr.py
@@ -2,7 +2,6 @@
 
 from typing import TYPE_CHECKING
 from typing import Any
-from typing import Callable
 from typing import Literal
 from typing import Sequence
 
@@ -16,6 +15,9 @@
 if TYPE_CHECKING:
     from typing_extensions import Self
 
+    from narwhals._compliant.typing import AliasNames
+    from narwhals._compliant.typing import EvalNames
+    from narwhals._compliant.typing import EvalSeries
     from narwhals._expression_parsing import ExprMetadata
     from narwhals._pandas_like.dataframe import PandasLikeDataFrame
     from narwhals._pandas_like.namespace import PandasLikeNamespace
@@ -71,12 +73,12 @@ def window_kwargs_to_pandas_equivalent(
 class PandasLikeExpr(EagerExpr["PandasLikeDataFrame", PandasLikeSeries]):
     def __init__(
         self: Self,
-        call: Callable[[PandasLikeDataFrame], Sequence[PandasLikeSeries]],
+        call: EvalSeries[PandasLikeDataFrame, PandasLikeSeries],
         *,
         depth: int,
         function_name: str,
-        evaluate_output_names: Callable[[PandasLikeDataFrame], Sequence[str]],
-        alias_output_names: Callable[[Sequence[str]], Sequence[str]] | None,
+        evaluate_output_names: EvalNames[PandasLikeDataFrame],
+        alias_output_names: AliasNames | None,
         implementation: Implementation,
         backend_version: tuple[int, ...],
         version: Version,
@@ -105,7 +107,7 @@ def __narwhals_expr__(self) -> None: ...
     @classmethod
     def from_column_names(
         cls: type[Self],
-        evaluate_column_names: Callable[[PandasLikeDataFrame], Sequence[str]],
+        evaluate_column_names: EvalNames[PandasLikeDataFrame],
         /,
         *,
         context: _FullContext,
diff --git a/narwhals/_spark_like/expr.py b/narwhals/_spark_like/expr.py
index 406c9b4db3..968a903d8d 100644
--- a/narwhals/_spark_like/expr.py
+++ b/narwhals/_spark_like/expr.py
@@ -30,6 +30,8 @@
     from typing_extensions import Self
 
     from narwhals._compliant.typing import AliasNames
+    from narwhals._compliant.typing import EvalNames
+    from narwhals._compliant.typing import EvalSeries
     from narwhals._expression_parsing import ExprMetadata
     from narwhals._spark_like.dataframe import SparkLikeLazyFrame
     from narwhals._spark_like.namespace import SparkLikeNamespace
@@ -42,10 +44,10 @@
 class SparkLikeExpr(LazyExpr["SparkLikeLazyFrame", "Column"]):
     def __init__(
         self: Self,
-        call: Callable[[SparkLikeLazyFrame], Sequence[Column]],
+        call: EvalSeries[SparkLikeLazyFrame, Column],
         *,
-        evaluate_output_names: Callable[[SparkLikeLazyFrame], Sequence[str]],
-        alias_output_names: Callable[[Sequence[str]], Sequence[str]] | None,
+        evaluate_output_names: EvalNames[SparkLikeLazyFrame],
+        alias_output_names: AliasNames | None,
         backend_version: tuple[int, ...],
         version: Version,
         implementation: Implementation,
@@ -119,10 +121,7 @@ def __narwhals_namespace__(self: Self) -> SparkLikeNamespace:  # pragma: no cove
             implementation=self._implementation,
         )
 
-    def _with_window_function(
-        self: Self,
-        window_function: WindowFunction,
-    ) -> Self:
+    def _with_window_function(self, window_function: WindowFunction) -> Self:
         result = self.__class__(
             self._call,
             evaluate_output_names=self._evaluate_output_names,
@@ -213,7 +212,7 @@ def func(window_inputs: WindowInputs) -> Column:
     @classmethod
     def from_column_names(
         cls: type[Self],
-        evaluate_column_names: Callable[[SparkLikeLazyFrame], Sequence[str]],
+        evaluate_column_names: EvalNames[SparkLikeLazyFrame],
         /,
         *,
         context: _FullContext,
diff --git a/narwhals/utils.py b/narwhals/utils.py
index b2024f3d7d..3d8a7dfae6 100644
--- a/narwhals/utils.py
+++ b/narwhals/utils.py
@@ -70,6 +70,7 @@
     from narwhals._compliant import CompliantSeriesT
     from narwhals._compliant import NativeFrameT_co
     from narwhals._compliant import NativeSeriesT_co
+    from narwhals._compliant.typing import EvalNames
     from narwhals._dask.namespace import DaskNamespace
     from narwhals._duckdb.namespace import DuckDBNamespace
     from narwhals._pandas_like.namespace import PandasLikeNamespace
@@ -1516,7 +1517,7 @@ def exclude_column_names(frame: _StoresColumns, names: Container[str]) -> Sequen
     return [col_name for col_name in frame.columns if col_name not in names]
 
 
-def passthrough_column_names(names: Sequence[str], /) -> Callable[[Any], Sequence[str]]:
+def passthrough_column_names(names: Sequence[str], /) -> EvalNames[Any]:
     def fn(_frame: Any, /) -> Sequence[str]:
         return names