Thanks, Minh.  That's the way I see it as well, though as usual you
are more eloquent.

So this is really a two-part question.  Should  lift()  for free
module morphisms be deprecated?

If so, in favor of

(a) a simple renaming, as perhaps "pre_image()"

OR

(b) make the existing  inverse_image()  method more capable:
submodule in, submodule out  plus  element in, element out.

At the risk of poisoning the well, I like (b), though in my textbook I
call this (a).

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