**Bool** \\( := (\mathbb{B}, \le, \tt{false}, \wedge) \\).

The following is not the complete proof, it only shows the mapping of data and
not that the rules of the two concepts are honored.

Given a preorder \\( \mathcal{P} : (P, \le) \\) use it to define, map to, a **Bool**-category \\( \mathcal{Q} \\).

The objects of \\( \mathcal{P} \\) are the objects of \\( \mathcal{Q} \\).

For every pair of objects \\( (x, y) \\) we assign an element of
\\( \mathbb{B} = \\\{false, true\\\} \\): simply assigning \\(\tt{true}\\) if \\(x \le y \\), and
\\(\tt{false} \\) otherwise.

Given a **Bool**-category \\( \mathcal{Q} \\) use it to define, map to, a preorder \\( \mathcal{P} \\).

The objects of \\( \mathcal{Q} \\) are the objects of \\( \mathcal{P} \\).

For every pair of objects \\( (x, y) \\) we observe the value of its element,
\\( \mathbb{B} = \\\{false, true\\\} \\): simply add the arrow in \\( \mathcal{P} \\)
when \\(\tt{true}\\) skipping the \\(\tt{false}\\) pairs.