> **Puzzle 171.** Is \$$\Phi(E,c) = \text{true}\$$ ?

**No** (if I am reading the diagram correctly)

> **Puzzle 172.** Is \$$\Phi(E,e) = \text{true}\$$?

**Yes**

We have \$$E \leq N\$$ so \$$\Phi(N,e) \leq \Phi(E,e)\$$. Since the diagram indicates \$$\Phi(N,e)=\text{true}\$$ then we know \$$\Phi(E,e) = \text{true}\$$.

> **Puzzle 175.** Suppose \$$f : X \to Y\$$ and \$$g : Y \to X\$$ are monotone functions, and use them to build feasibility relations \$$\Phi\$$ and \$$\Psi\$$ as in the previous two puzzles. When is
>
> $\Phi = \Psi ?$

When \$$f\$$ and \$$g\$$ are adjoints, that is to say \$$f \dashv g\$$