**Galois connection a.k.a. Adjunction Formula**
\[
f(p) \le q \iff p \le g(q) \\
\text{Where g is right-adjoint to f and f is left-adjoint to g}
\]

1. Choose a nontrivial partition \\( c : S \twoheadrightarrow P \\) and let \\( g_!(c) \\) be its push forward partition on T.
2. Choose any coarser partition \\( d : T \twoheadrightarrow P' \\), i.e. where \\( g_!(c) \le d \\) .
3. Choose any non-coarser partition \\( e : T \twoheadrightarrow Q \\), i.e. where \\( g_!(c) \nleq e \\). (If you can’t do this, revise your answer for #1.)
4. Find \\( g^\*(d) \text{ and } g^\*(e) \\) .
5. The adjunction formula, in this case, says that since \\( g_!(c) \le d \text{ and } g_!(c) \nleq e \\) , we should have \\( c \le g^\*(d) \text{ and } c \nleq g^\*(e) \\) . Show that this is true.

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