Speaking of analogies, I had an insight regarding the relation between profunctors and electrical circuits, that helped me to intuit Dan's answer to the puzzle. Each profunctor (or tensored pair of profunctors) can be modeled by a circuit, and circuits compose. \$$\mathcal{V}\$$-enrichment means we can "measure" a value in \$$\mathcal{V}\$$ for objects in a category (the hom-functor), and profunctors allow us to measure those values across (tensored) categories. As Anindya said, this is like creating a bridge between categories - just as we can measure current between any two nodes in a circuit (even if the two nodes belong to different circuits themselves).

Consider also the principle that the flow of current is inversely proportional to resistance (i.e. Ohm's law https://en.wikipedia.org/wiki/Ohm%27s_law]). This seems related to the quantale nature of \$$\mathbf{Prof}_\mathcal{V}\$$, namely the path of least resistance is a join. I'm certain this has been elaborated much further here: https://arxiv.org/abs/1504.05625

This leads to a formulation of the puzzle as: prove that the snake equation forms a circuit whose end-to-end value in \$$\mathcal{V}\$$ is the same as measuring the value across the identity profunctor. Something like the following: