Can we give the \$$\mathbf{Bool}\$$-companion as a natural transformation:

\$\alpha\_⤴ := \begin{matrix} & & \hat{F} & & \\\\ & X & \nrightarrow & Y & \\\\ F & \downarrow & & \downarrow & Id\_Y\\\\ & Y & \rightarrow & Y & \\\\ & & \leq\_Y & & \end{matrix}\quad ? \$

Edit: From what I understand from [here](https://golem.ph.utexas.edu/category/2017/08/a_graphical_calculus_for_proar.html), we should also have another natural transformation for the companion. I believe,

\$\alpha\_↱ := \begin{matrix} & & \leq\_X & & \\\\ & X & \rightarrow & X & \\\\ Id\_X & \downarrow & & \downarrow & F\\\\ & X & \nrightarrow & Y. & \\\\ & & \hat{F} & & \end{matrix} \$