**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.