diff --git a/src/lightning/pytorch/trainer/connectors/accelerator_connector.py b/src/lightning/pytorch/trainer/connectors/accelerator_connector.py
index 82da5248e2d6d..7f44de0589938 100644
--- a/src/lightning/pytorch/trainer/connectors/accelerator_connector.py
+++ b/src/lightning/pytorch/trainer/connectors/accelerator_connector.py
@@ -453,10 +453,11 @@ def _check_strategy_and_fallback(self) -> None:
 
         if (
             strategy_flag in FSDPStrategy.get_registered_strategies() or type(self._strategy_flag) is FSDPStrategy
-        ) and self._accelerator_flag not in ("cuda", "gpu"):
+        ) and not (self._accelerator_flag in ("cuda", "gpu") or isinstance(self._accelerator_flag, CUDAAccelerator)):
             raise ValueError(
-                f"The strategy `{FSDPStrategy.strategy_name}` requires a GPU accelerator, but got:"
-                f" {self._accelerator_flag}"
+                f"The strategy `{FSDPStrategy.strategy_name}` requires a GPU accelerator, but received "
+                f"`accelerator={self._accelerator_flag!r}`. Please set `accelerator='cuda'`, `accelerator='gpu'`,"
+                " or pass a `CUDAAccelerator()` instance to use FSDP."
             )
         if strategy_flag in _DDP_FORK_ALIASES and "fork" not in torch.multiprocessing.get_all_start_methods():
             raise ValueError(
diff --git a/tests/tests_pytorch/trainer/connectors/test_accelerator_connector.py b/tests/tests_pytorch/trainer/connectors/test_accelerator_connector.py
index d079a1c7b9a1e..f3d98cf444c36 100644
--- a/tests/tests_pytorch/trainer/connectors/test_accelerator_connector.py
+++ b/tests/tests_pytorch/trainer/connectors/test_accelerator_connector.py
@@ -582,6 +582,11 @@ class AcceleratorSubclass(CPUAccelerator):
     Trainer(accelerator=AcceleratorSubclass(), strategy=FSDPStrategySubclass())
 
 
+@RunIf(min_cuda_gpus=1)
+def test_check_fsdp_strategy_and_fallback_with_cudaaccelerator():
+    Trainer(strategy="fsdp", accelerator=CUDAAccelerator())
+
+
 @mock.patch.dict(os.environ, {}, clear=True)
 def test_unsupported_tpu_choice(xla_available, tpu_available):
     # if user didn't set strategy, _Connector will choose the SingleDeviceXLAStrategy or XLAStrategy