I'm still getting my head around this myself, but I know a few
references that might help (maybe not directly, but they're in the
ballpark).

1 I believe the phrase "natural lifting" or "naturality of liftings"
is what you're after when you attempt to compare a monad and a "bigger
monad" that includes the affects of the first. I recall this concept
from Liang and Hudak's modular monadic semantics work, but I'm having
a heck of a time quickly finding it in their papers. Try at least
section 8.1 in Monad Transformers and Modular Interpreters.

2 Another example that helped me when getting a feel for reasoning
about monadic code (which is the basis of what we're doing here) was
William Harrison's "Proof Abstraction for Imperative Languages
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