Christopher wrote:

> I think of them as relation like, because relations have exactly that same orientation but not direction.

Profunctors are indeed similar to relations, but they have more of a directionality than relations.

A relation between two sets

$R \colon X \nrightarrow Y$

can be seen as a function

$R \colon X \times Y \to \mathbf{Bool}$

where \$$R(x,y) = \texttt{true}\$$ if and only if the relation _holds_ between \$$x\$$ and \$$y\$$.

A \$$\mathcal{V}\$$-enriched profunctor between two \$$\mathcal{V}\$$-enriched categories

$\Phi \colon \mathcal{X} \nrightarrow \mathcal{Y}$

can be seen as an enriched functor

$\Phi \colon \mathcal{X}^{\text{op}} \times \mathcal{Y} \to \mathcal{V} .$

Notice that 'op'. That makes a difference!

The difference is that a relation

$R \colon X \nrightarrow Y$

can always be 'turned around' to give a relation going back the other way, called the **[transpose](https://en.wikipedia.org/wiki/Converse_relation)** and written

$R^\top \colon Y \nrightarrow X,$

while a profunctor

$\Phi \colon \mathcal{X} \nrightarrow \mathcal{Y}$

_cannot_ be turned around to give a profunctor from \$$\mathcal{Y}\$$ to \$$\mathcal{X}\$$... only one from \$$\mathcal{Y}^{\text{op}}\$$ to \$$\mathcal{X}^{\text{op}}\$$.

If we think of a relation as a function

$R \colon X \times Y \to \mathbf{Bool}$

then its transpose is the function

$R^\top \colon Y \times X \to \mathbf{Bool}$

given by

$R^\top(y,x) = R(x,y)$

It's just like the transpose of a matrix, and indeed it's good to visualize a relation as an \$$X \times Y\$$-shaped box of \$$\texttt{true}\$$s and \$$\texttt{false}\$$s.

If we think of an enriched profunctor as a enriched functor

$\Phi \colon \mathcal{X}^{\text{op}} \times \mathcal{Y} \to \mathcal{V} .$

then its transpose is

$\Phi^\top \colon (\mathcal{Y}^{\text{op}})^{\text{op}} \times \mathcal{X}^\text{op} \to \mathcal{V}$

given by

$\Phi^\top(y,x) = \Phi(x,y)$

But note, this gives a profunctor

$\Phi^{\top} \colon \mathcal{X}^{\text{op}} \to \mathcal{Y}^{\text{op}}$

So with profunctors, you have to 'flip them upside down as you turn them around'.