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.

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.