diff --git a/advanced_source/cpp_custom_ops.rst b/advanced_source/cpp_custom_ops.rst
index 435ff088bc..ffabd6eff7 100644
--- a/advanced_source/cpp_custom_ops.rst
+++ b/advanced_source/cpp_custom_ops.rst
@@ -174,6 +174,8 @@ To add ``torch.compile`` support for an operator, we must add a FakeTensor kerne
 known as a "meta kernel" or "abstract impl"). FakeTensors are Tensors that have
 metadata (such as shape, dtype, device) but no data: the FakeTensor kernel for an
 operator specifies how to compute the metadata of output tensors given the metadata of input tensors.
+The FakeTensor kernel should return dummy Tensors of your choice with
+the correct Tensor metadata (shape/strides/``dtype``/device).
 
 We recommend that this be done from Python via the `torch.library.register_fake` API,
 though it is possible to do this from C++ as well (see
diff --git a/advanced_source/python_custom_ops.py b/advanced_source/python_custom_ops.py
index 1e429b76b3..0b3bf6e474 100644
--- a/advanced_source/python_custom_ops.py
+++ b/advanced_source/python_custom_ops.py
@@ -66,7 +66,7 @@ def display(img):
 ######################################################################
 # ``crop`` is not handled effectively out-of-the-box by
 # ``torch.compile``: ``torch.compile`` induces a
-# `"graph break" <https://pytorch.org/docs/stable/torch.compiler_faq.html#graph-breaks>`_ 
+# `"graph break" <https://pytorch.org/docs/stable/torch.compiler_faq.html#graph-breaks>`_
 # on functions it is unable to handle and graph breaks are bad for performance.
 # The following code demonstrates this by raising an error
 # (``torch.compile`` with ``fullgraph=True`` raises an error if a
@@ -85,9 +85,9 @@ def f(img):
 #
 # 1. wrap the function into a PyTorch custom operator.
 # 2. add a "``FakeTensor`` kernel" (aka "meta kernel") to the operator.
-#    Given the metadata (e.g. shapes)
-#    of the input Tensors, this function says how to compute the metadata
-#    of the output Tensor(s).
+#    Given some ``FakeTensors`` inputs (dummy Tensors that don't have storage),
+#    this function should return dummy Tensors of your choice with the correct
+#    Tensor metadata (shape/strides/``dtype``/device).
 
 
 from typing import Sequence
@@ -130,6 +130,11 @@ def f(img):
 # ``autograd.Function`` with PyTorch operator registration APIs can lead to (and
 # has led to) silent incorrectness when composed with ``torch.compile``.
 #
+# If you don't need training support, there is no need to use
+# ``torch.library.register_autograd``.
+# If you end up training with a ``custom_op`` that doesn't have an autograd
+# registration, we'll raise an error message.
+#
 # The gradient formula for ``crop`` is essentially ``PIL.paste`` (we'll leave the
 # derivation as an exercise to the reader). Let's first wrap ``paste`` into a
 # custom operator:
@@ -203,7 +208,7 @@ def setup_context(ctx, inputs, output):
 ######################################################################
 # Mutable Python Custom operators
 # -------------------------------
-# You can also wrap a Python function that mutates its inputs into a custom 
+# You can also wrap a Python function that mutates its inputs into a custom
 # operator.
 # Functions that mutate inputs are common because that is how many low-level
 # kernels are written; for example, a kernel that computes ``sin`` may take in