**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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