I could contribute how I understand **Puzzle TR1**, though I'm not sure how terse it is.

We know that \$$g(b)\$$ is an upper bound for the set \$$A_b = \\{a \in A : f(a) \leq_B b\\}\$$. We want to show that \$$g(b)\$$ is a *least* upper bound for \$$A_b\$$.

It is enough to show that \$$g(b) \in A_b\$$, from which it follows that any upper bound for \$$A_b\$$ must be at least \$$g(b)\$$.

But \$$g(b) \in A_b\$$ iff \$$f(g(b)) \leq_B b\$$ [by the definition \$$A_b\$$], iff \$$g(b) \leq_A g(b)\$$ [by the definition of adjoints], which is true.