diff --git a/python-package/lightgbm/engine.py b/python-package/lightgbm/engine.py
index 3a0c93fba332..ca5428250b8f 100644
--- a/python-package/lightgbm/engine.py
+++ b/python-package/lightgbm/engine.py
@@ -11,7 +11,7 @@
 
 from . import callback
 from .basic import (Booster, Dataset, LightGBMError, _choose_param_value, _ConfigAliases, _InnerPredictor,
-                    _LGBM_CategoricalFeatureConfiguration, _LGBM_CustomObjectiveFunction,
+                    _LGBM_CategoricalFeatureConfiguration, _LGBM_CustomObjectiveFunction, _LGBM_EvalFunctionResultType,
                     _LGBM_FeatureNameConfiguration, _log_warning)
 from .compat import SKLEARN_INSTALLED, _LGBMBaseCrossValidator, _LGBMGroupKFold, _LGBMStratifiedKFold
 
@@ -22,9 +22,15 @@
 ]
 
 
-_LGBM_CustomMetricFunction = Callable[
-    [np.ndarray, Dataset],
-    Union[Tuple[str, float, bool], List[Tuple[str, float, bool]]]
+_LGBM_CustomMetricFunction = Union[
+    Callable[
+        [np.ndarray, Dataset],
+        _LGBM_EvalFunctionResultType,
+    ],
+    Callable[
+        [np.ndarray, Dataset],
+        List[_LGBM_EvalFunctionResultType]
+    ],
 ]
 
 _LGBM_PreprocFunction = Callable[
diff --git a/python-package/lightgbm/sklearn.py b/python-package/lightgbm/sklearn.py
index e5186415783e..8f42a6c47400 100644
--- a/python-package/lightgbm/sklearn.py
+++ b/python-package/lightgbm/sklearn.py
@@ -33,32 +33,50 @@
     scipy.sparse.spmatrix
 ]
 _LGBM_ScikitCustomObjectiveFunction = Union[
+    # f(labels, preds)
     Callable[
-        [np.ndarray, np.ndarray],
+        [Optional[np.ndarray], np.ndarray],
         Tuple[np.ndarray, np.ndarray]
     ],
+    # f(labels, preds, weights)
     Callable[
-        [np.ndarray, np.ndarray, np.ndarray],
+        [Optional[np.ndarray], np.ndarray, Optional[np.ndarray]],
         Tuple[np.ndarray, np.ndarray]
     ],
+    # f(labels, preds, weights, group)
     Callable[
-        [np.ndarray, np.ndarray, np.ndarray, np.ndarray],
+        [Optional[np.ndarray], np.ndarray, Optional[np.ndarray], Optional[np.ndarray]],
         Tuple[np.ndarray, np.ndarray]
     ],
 ]
 _LGBM_ScikitCustomEvalFunction = Union[
+    # f(labels, preds)
     Callable[
-        [np.ndarray, np.ndarray],
-        Union[_LGBM_EvalFunctionResultType, List[_LGBM_EvalFunctionResultType]]
+        [Optional[np.ndarray], np.ndarray],
+        _LGBM_EvalFunctionResultType
     ],
     Callable[
-        [np.ndarray, np.ndarray, np.ndarray],
-        Union[_LGBM_EvalFunctionResultType, List[_LGBM_EvalFunctionResultType]]
+        [Optional[np.ndarray], np.ndarray],
+        List[_LGBM_EvalFunctionResultType]
     ],
+    # f(labels, preds, weights)
     Callable[
-        [np.ndarray, np.ndarray, np.ndarray, np.ndarray],
-        Union[_LGBM_EvalFunctionResultType, List[_LGBM_EvalFunctionResultType]]
+        [Optional[np.ndarray], np.ndarray, Optional[np.ndarray]],
+        _LGBM_EvalFunctionResultType
     ],
+    Callable[
+        [Optional[np.ndarray], np.ndarray, Optional[np.ndarray]],
+        List[_LGBM_EvalFunctionResultType]
+    ],
+    # f(labels, preds, weights, group)
+    Callable[
+        [Optional[np.ndarray], np.ndarray, Optional[np.ndarray], Optional[np.ndarray]],
+        _LGBM_EvalFunctionResultType
+    ],
+    Callable[
+        [Optional[np.ndarray], np.ndarray, Optional[np.ndarray], Optional[np.ndarray]],
+        List[_LGBM_EvalFunctionResultType]
+    ]
 ]
 _LGBM_ScikitEvalMetricType = Union[
     str,
@@ -135,11 +153,11 @@ def __call__(self, preds: np.ndarray, dataset: Dataset) -> Tuple[np.ndarray, np.
         labels = dataset.get_label()
         argc = len(signature(self.func).parameters)
         if argc == 2:
-            grad, hess = self.func(labels, preds)
+            grad, hess = self.func(labels, preds)  # type: ignore[call-arg]
         elif argc == 3:
-            grad, hess = self.func(labels, preds, dataset.get_weight())
+            grad, hess = self.func(labels, preds, dataset.get_weight())  # type: ignore[call-arg]
         elif argc == 4:
-            grad, hess = self.func(labels, preds, dataset.get_weight(), dataset.get_group())
+            grad, hess = self.func(labels, preds, dataset.get_weight(), dataset.get_group())  # type: ignore [call-arg]
         else:
             raise TypeError(f"Self-defined objective function should have 2, 3 or 4 arguments, got {argc}")
         return grad, hess
@@ -213,11 +231,11 @@ def __call__(
         labels = dataset.get_label()
         argc = len(signature(self.func).parameters)
         if argc == 2:
-            return self.func(labels, preds)
+            return self.func(labels, preds)  # type: ignore[call-arg]
         elif argc == 3:
-            return self.func(labels, preds, dataset.get_weight())
+            return self.func(labels, preds, dataset.get_weight())  # type: ignore[call-arg]
         elif argc == 4:
-            return self.func(labels, preds, dataset.get_weight(), dataset.get_group())
+            return self.func(labels, preds, dataset.get_weight(), dataset.get_group())  # type: ignore[call-arg]
         else:
             raise TypeError(f"Self-defined eval function should have 2, 3 or 4 arguments, got {argc}")
 
@@ -819,7 +837,7 @@ def _get_meta_data(collection, name, i):
             num_boost_round=self.n_estimators,
             valid_sets=valid_sets,
             valid_names=eval_names,
-            feval=eval_metrics_callable,
+            feval=eval_metrics_callable,  # type: ignore[arg-type]
             init_model=init_model,
             feature_name=feature_name,
             callbacks=callbacks