Let’s consider Bool-profunctors. Recall a preorder (Bool-category) can be drawn as a Hasse diagram.
A Bool-profunctor \(X \overset{\phi}{\nrightarrow} Y\) can look like this:
With bridges coming from the profunctor, one can now use both paths to get from points in \(X\) to points in \(Y\).
There is a path from N to e, so \(\phi(N,e)=\)\(true\) but \(\phi(W,d)=\)\(false\).
We could put a box around both preorders and define a new preorder, called the collage.