One of the things I've found slightly tricky about profunctors is avoiding the temptation to imagine \\(\Phi : \mathcal{X} \nrightarrow \mathcal{Y}\\) as some sort of map taking things in \\(\mathcal{X}\\) to things in \\(\mathcal{Y}\\). This picture kicks in almost automatically as soon as I see an arrow in a category, but it is really misleading in this case.

In fact I'd go further and suggest it's a bad idea to think of \\(\Phi : \mathcal{X} \nrightarrow \mathcal{Y}\\) as "going from \\(\mathcal{X}\\) to \\(\mathcal{Y}\\)" in any sense. The arrow is a label telling you which way the profunctor is **oriented** rather than telling you which **direction** in "goes in".

What I'm getting at here is that "orientation" is looser than "direction", in that it's an arbitrary choice based on convention and convenience rather than something that refers to an essential property of the profunctor. We *could* have picked the profunctor \\(\Phi^\text{op} : \mathcal{Y}^\text{op} \nrightarrow \mathcal{X}^\text{op}\\) instead. That really isn't the case with a function from \\(X\\) to \\(Y\\) which absolutely is not any kind of function the other way round.

In terms of *how* to picture profunctors, if not as directed arrows, I'm tending to think that \\(\Phi : \mathcal{X} \nrightarrow \mathcal{Y}\\) should be seen as a kind of **bridge** from \\(\mathcal{X}\\) to \\(\mathcal{Y}\\). The intuition here is that you can "cross" in either direction, or even just stand in the middle and admire the scenery. And "composing" profunctors is not composition in the sense of "do this, *then* do this, *then* do this..." – rather it's more a case of **connecting** bridges to build bigger bridges, without any of the temporal connotations involved when we compose functions.

In fact I'd go further and suggest it's a bad idea to think of \\(\Phi : \mathcal{X} \nrightarrow \mathcal{Y}\\) as "going from \\(\mathcal{X}\\) to \\(\mathcal{Y}\\)" in any sense. The arrow is a label telling you which way the profunctor is **oriented** rather than telling you which **direction** in "goes in".

What I'm getting at here is that "orientation" is looser than "direction", in that it's an arbitrary choice based on convention and convenience rather than something that refers to an essential property of the profunctor. We *could* have picked the profunctor \\(\Phi^\text{op} : \mathcal{Y}^\text{op} \nrightarrow \mathcal{X}^\text{op}\\) instead. That really isn't the case with a function from \\(X\\) to \\(Y\\) which absolutely is not any kind of function the other way round.

In terms of *how* to picture profunctors, if not as directed arrows, I'm tending to think that \\(\Phi : \mathcal{X} \nrightarrow \mathcal{Y}\\) should be seen as a kind of **bridge** from \\(\mathcal{X}\\) to \\(\mathcal{Y}\\). The intuition here is that you can "cross" in either direction, or even just stand in the middle and admire the scenery. And "composing" profunctors is not composition in the sense of "do this, *then* do this, *then* do this..." – rather it's more a case of **connecting** bridges to build bigger bridges, without any of the temporal connotations involved when we compose functions.