> 2) Write down a profunctor \$$\Lambda : \mathcal{X} \nrightarrow \mathcal{Y} \$$ and, reading \$$\Lambda(x, y) = \text{true} \$$ as
“my uncle can explain \$$x\$$ given \$$y\$$”, give an interpretation of the fact that the preimage of
\$$\text{true} \$$ forms an upper set in \$$\mathcal{X}^{op} \times \mathcal{Y} \$$.

I am not entirely sure what it means to "write down a profunctor" but I will make a guess.

$\begin{array}{c c | c} x & y & p \\\\ \text{monoid} & \text{this book} & \text{true} \\\\ \text{monoid} & \text{nothing} & \text{false} \\\\ \text{preorder} & \text{this book} & \text{true} \\\\ \text{preorder} & \text{nothing} & \text{false} \\\\ \text{category} & \text{this book} & \text{true} \\\\ \text{category} & \text{nothing} & \text{false} \end{array}$

My uncle can explain categories, monoids and preorders if he has studied the book.

As will be seen later this can also be expressed as...

$\begin{array}{c c | c} \Lambda & \text{this book} & \text{nothing} \\\\ \text{monoid} & \text{true} & \text{false} \\\\ \text{preorder} & \text{true} & \text{false} \\\\ \text{category} & \text{true} & \text{false} \end{array}$
Another way to think about this is to convert the \$$\le \$$ into "better".
Converting the consuming side means \$$X_{op} \$$.