Fix a field.  Given a vector space $V$ it has a dual $V^\ast$, which has its own dual $V^{\ast \ast}$, which has its own dual $V^{\ast \ast \ast}$, and so on *ad infinitum*.

There are lots of natural maps between these iterated duals.  Most famous is the map

$$ i_V \colon V \to V^{\ast \ast} $$

given by

$$ i_V(v)(f) = f(v)  \quad \forall v \in V, f \in V^\ast $$

but there are many others.  For example, we have the map

$$ i_{V^{\ast \ast}} \colon V^{\ast \ast} \to V^{\ast \ast \ast \ast} $$

but we also have the map 

$$ i_{V^{\ast}}^\ast \colon V^{\ast \ast \ast \ast} \to V^{\ast \ast}$$

formed by taking the adjoint of 

$$ i_{V^{\ast}} \colon V^{\ast} \to V^{\ast \ast \ast}. $$

And we have various equations between these maps.  For example, I believe $i_{V^{\ast}}^\ast$ is a left inverse of $i_{V^{\ast \ast}}$:

$$ i_{V^{\ast}}^\ast \circ i_{V^{\ast \ast}} = 1. $$

So it's natural to want to get to the bottom of this and ask what are *all* the natural maps between iterated duals, and *all* the equations involving these maps.

To formalize this it's good to treat duality as a pair of contravariant functors

$$ D \colon \mathrm{Vect} \to \mathrm{Vect}^{\rm op}$$ 

$$ E \colon \mathrm{Vect}^{\rm op} \to \mathrm{Vect}.$$ 

These are really the same functor in two disguises: namely, the contravariant functor that sends any vector space $V$ to its dual $V^\ast$, and any linear map $f\colon V \to W$ to its adjoint $f^{\ast} \colon W^\ast \to V^\ast$.   Treating them as two separate functors makes it a bit easier to make sense of the fact that they are adjoint functors and thus [they define a monad](https://mathoverflow.net/questions/396182/the-bidualizing-monad) and also a comonad.  

A bunch of natural transformations between the powers of 

$$ \begin{array}{cccc} E \circ D \colon & \mathrm{Vect} &\to& \mathrm{Vect}  \\
& V &\mapsto& V^{\ast \ast} ,
\end{array}
$$

and also a bunch of equations between these natural transformations, 
arise from the adjunction between $D$ and $E$.   So, one can ask if *all* the natural transformations between the powers of $E \circ D$, and *all* the equations between these, arise from this adjunction.

To give this conjecture a chance to be true, we need to interpret 'arise from' broadly enough.  For example, a linear combination of natural transformations will again be a natural transformation.  So, to have a chance of getting all the natural transformations, we need to start with the unit and counit of the adjunction, and build other natural transformations from these, and the functors $D$ and $E$, by taking all composites (in the 2-categorical sense), and also linear combinations.   Then we can ask whether the resulting natural transformations are *all* the natural transformations between powers of $E \circ D$.   

We can also ask whether all equations between these natural transformations follow from the usual 'zig-zag equations' governing the unit and counit of an adjunction... where 'follow' needs to be made more precise.

Ultimately the slickest way to formulate the question may be something like this.  There is a 2-category $\mathbf{Adj}$ called the 'walking adjunction', such that an adjunction in $\mathbf{Cat}$ is the same as a 2-functor 

$$ F \colon \mathbf{Adj} \to \mathbf{Cat} $$

The adjunction between $D$ and $E$ thus gives a 2-functor $F \colon \mathbf{Adj} \to \mathbf{Cat}$, and it should be possible to express some version of my question as a question about this 2-functor.  For example, we can ask if this 2-functor is **locally faithful**, meaning faithful on each hom-category.   If it's not, there are additional equations involving the unit and counit of this particular adjunction, that don't hold in a general adjunction.   

However, if additional natural maps are known between iterated duals, or additional equations between these, someone may have discovered them without phrasing it in terms of 2-categories, or even categories.

I should add that everything changes if we restrict to finite-dimensional vector spaces, because then $i_V$ is an isomorphism.  I *don't* want to restrict to finite-dimensional vector spaces.