Just to elaborate and add a bit of beginner's background to Michael's answer to Puzzle 283, from someone who didn't know what dual spaces were and had to look them up:

The dual space has as elements "linear functionals", or equivalently covectors, \$$\phi \colon V \rightarrow \mathcal{k}\$$. These can be thought of as morphisms that apply a row vector to a column vector to yield a real number value. In other words, it takes a vector representing the values of a linear polynomial and applies coefficients to get a real number. So the tensored category \$$V \otimes V^\ast\$$ contains tuples \$$\langle v, \phi \rangle\$$ of vectors and linear functionals. \$$\cap_V\$$ is a profunctor between \$$k\$$ and \$$V \otimes V^\ast\$$. That's like asking about the feasibility of a real number being equal to the value of a vector applied to a linear functional. Namely, a real number \$$n\$$ is related to a tuple \$$\langle v, \phi \rangle\$$ iff \$$\phi(v) = n\$$.

(As an aside, it seems arbitrary that we use \$$=\$$ here instead of \$$\leq\$$. We could move from enriching our relations with \$$(\mathcal{R}, =, 1, *$$\\) to enriching them in \$$(\mathcal{R}, \leq, 1, *)\$$, so that we're asking about the feasibility of a real number being less than or equal to \$$\phi(v)\$$. Not sure if that makes sense though.)

So to translate this into the snake equation, we:

1. take any vector \$$v\$$ and factor out some \$$k\$$ - because we're allowed to do that with scalars and vectors. now we have a tuple \$$\langle k, 1/k * v \rangle\$$.

2. transform (or coevaluate) the \$$k\$$ into \$$\langle v_k, \phi_k \rangle\$$ via the \$$\cap_V\$$ - since it's "feasible" to do. notice that in parallel we're also taking \$$1/k * v\$$ to itself.

3. do the associator magic which takes every \$$\langle\langle v_k, \phi_k \rangle, 1/k * v \rangle \$$ to \$$\langle v_k, \langle \phi_j, 1/k * v \rangle\rangle\$$.

4. transform (or evaluate) \$$\langle \phi_j, 1/k * v \rangle\$$ to some new real number \$$j\$$, and in parallel take \$$v_k\$$ to \$$v_k\$$.

5. multiply \$$v_k\$$ by \$$j\$$ to get your original vector \$$v\$$.

In addition, we need to show that for any input \$$v\$$ we can choose some \$$k, v_k, \phi_k, \phi_j, j\$$ that will make this work. Well, just let \$$k\$$ and \$$j\$$ be reciprocals, and let \$$\phi_j(1/k * v) = j = 1/k\$$. Notice that the associator doesn't restrict us to using the same linear functional - we can pick whatever we want to make this equation work so long as it's 1:1.