Sorry, David: a profunctor typically does not give rise to a function on objects. In particular,

\[ \cap_{\mathcal{X}} (1) = x \otimes x \]

does not make sense. And there's another way to see something is wrong here: what's \\(x\\) supposed to be on the right-hand side? There's no \\(x\\) on the left-hand side!

A profunctor that's the companion of a functor _does_ gives rise to a function on objects. An example would be the associator. But the cap and the cup are not companions of anything.

I think I've failed to adequately explain how to compose profunctors. I gave the formula in [Lecture 63](https://forum.azimuthproject.org/discussion/2295/lecture-63-chapter-4-composing-enriched-profunctors/p1), and people solved some puzzles based on this, but when it comes to more complicated applications, like Puzzle 227 in this lecture, people seem to be stuck.

I'll start by reminding everyone of the formula:

> **Definition.** If \\(\mathcal{V}\\) is a commutative quantale and \\(\Phi \colon \mathcal{X} \nrightarrow \mathcal{Y}\\), \\(\Psi\colon \mathcal{Y} \nrightarrow \mathcal{Z}\\) are \\(\mathcal{V}\\)-enriched profunctors, define their **composite** by

> \[ (\Psi\Phi)(x,z) = \bigvee_{y \in \mathrm{Ob}(\mathcal{Y})} \Phi(x,y) \otimes \Psi(y,z). \]

So, profunctors are like matrices, and the formula for composing them is like matrix multiplication, with \\(\otimes\\) replacing multiplication, and \\(\bigvee\\) replacing \\(\sum\\). Remember, matrix multiplication goes like this:

\[ (AB)\_{i k} = \sum\_j A_\{i j} B\_{j k} .\]

It's the same basic idea.

\[ \cap_{\mathcal{X}} (1) = x \otimes x \]

does not make sense. And there's another way to see something is wrong here: what's \\(x\\) supposed to be on the right-hand side? There's no \\(x\\) on the left-hand side!

A profunctor that's the companion of a functor _does_ gives rise to a function on objects. An example would be the associator. But the cap and the cup are not companions of anything.

I think I've failed to adequately explain how to compose profunctors. I gave the formula in [Lecture 63](https://forum.azimuthproject.org/discussion/2295/lecture-63-chapter-4-composing-enriched-profunctors/p1), and people solved some puzzles based on this, but when it comes to more complicated applications, like Puzzle 227 in this lecture, people seem to be stuck.

I'll start by reminding everyone of the formula:

> **Definition.** If \\(\mathcal{V}\\) is a commutative quantale and \\(\Phi \colon \mathcal{X} \nrightarrow \mathcal{Y}\\), \\(\Psi\colon \mathcal{Y} \nrightarrow \mathcal{Z}\\) are \\(\mathcal{V}\\)-enriched profunctors, define their **composite** by

> \[ (\Psi\Phi)(x,z) = \bigvee_{y \in \mathrm{Ob}(\mathcal{Y})} \Phi(x,y) \otimes \Psi(y,z). \]

So, profunctors are like matrices, and the formula for composing them is like matrix multiplication, with \\(\otimes\\) replacing multiplication, and \\(\bigvee\\) replacing \\(\sum\\). Remember, matrix multiplication goes like this:

\[ (AB)\_{i k} = \sum\_j A_\{i j} B\_{j k} .\]

It's the same basic idea.