[Bartosz Milewski](https://bartoszmilewski.com/2017/07/07/profunctor-optics-the-categorical-view/) gives composition of profunctors as a coend,

\$(\Phi\circ\Psi)(x,z) = \int^{y\text{ in }Y}\Phi(x,y)\otimes\Psi(y,z), \$

but for \$$Bool\$$-profunctors we have

\$(\Phi\circ\Psi)(x,z) = \exists_{y\text{ in }Y}.\Phi(x,y) \land \Psi(y,z), \$

and also above we've also seen that composition of profunctors this could also be seen as matrix multiplication,

\$(\Phi\circ\Psi)(x,z) = \sum_{y\text{ in }Y}\Phi_x^y \times \Psi_y^z . \$

Are coends (\$$\int^c\$$), sums (\$$\sum_c\$$), and existential quantifiers (\$$\exists_c\$$) all related in some way ( I suspect they're all examples of folds of some sort)?