We can express \$$\Phi\$$ as a matrix where the \$$(m, n)^{th} \$$
entry is the value of \$$\Phi(m, n) \in \mathbb{B} \$$ .
Fill out the \$$\textbf{Bool}\$$-matrix:

$\begin{array}{c | c c c c c} \Phi & a & b & c & d & e \\\\ \hline N & ? & ? & ? & ? & true \\\\ E & true & ? & ? & ? & ? \\\\ W & ? & ? & ? & false & ? \\\\ S & ? & ? & ? & ? & ? \end{array}$

We’ll call this the feasibility matrix of \$$\Phi\$$.

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$\textbf{Bool}\text{-profunctor } \Phi : X \nrightarrow Y$