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}
\\]