[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)?

\\[

(\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)?