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You've lost track of who's on which side. Heat does not flow.
Heat is
the name of any process by means of which energy is transferred from
one system to another with no resulting change in external coordinates
of either system. Work is the name we give to any process by which
energy is transferred from one system to another by means of changing
the external coordinates when the systems are themally isolated from
one another.
We say "energy flows" from one system to another;
we never
say "work flows", and we *should not* say "heat flows".
That is the
conventional assignment of meaning to those words. If you do not wish
to use those terms that is your choice, but you do students a great
disservice by teaching them language that other physicists do not use.
2) Can you tell me how it is that the cool object increases in energy with
out saying the words "heat" and "flow"? What is the detailed mechanism?
Say that energy flows.
Who cares what the "detailed" mechanism is?