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

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

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

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

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

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 from $\mathbf{Adj}$ to $\mathbf{Cat}$.

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

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

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

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

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

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 from $\mathbf{Adj}$ to $\mathbf{Cat}$.

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

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

$$F_k \colon \mathbf{Adj}_k \to \mathbf{Cat}_k,$$

Question 2. Is $F_k \colon \mathbf{Adj}_k \to \mathbf{Cat}_k$ locally full and locally faithtful?

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

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

$$F_k \colon \mathbf{Adj}_k \to \mathbf{Cat}_k$$

Question 2. Is $F_k \colon \mathbf{Adj}_k \to \mathbf{Cat}_k$ locally full and locally faithful?

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.

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.

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.

The slickest way to formulate the question may use a bit of 2-category theory.

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 from $\mathbf{Adj}$ to $\mathbf{Cat}$.

The adjunction between $D$ and $E$ thus gives a particular 2-functor

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

and we can express part of my question as a question about this 2-functor. For example, we can ask

Question 1. Is $F \colon \mathbf{Adj} \to \mathbf{Cat}$ locally faithful, meaning faithful on each hom-category?

If it's not, there are additional equations involving the unit and counit of the adjunction between $D$ and $E$, that don't hold in a general adjunction.

To ask whether we've found all the natural maps between iterated duals of vector spaces, we might ask if $F$ is locally full, meaning full on each hom-category. But we already know it's not, due the linearity issue I mentioned! So it's good to follow Peter LeFanu Lumsdaine's suggestions and work instead with something like $\mathbf{Adj}_k$, the walking additive $k$-linear adjunction. This is a locally additive $k$-linear 2-category such that a 2-functor into $\mathbf{Cat}_k$, the 2-category of additive $k$-linear categories, is the same as an adjunction in $\mathbf{Cat}_k$.

The adjunction between $D$ and $E$ thus gives a particular 2-functor

$$F_k \colon \mathbf{Adj}_k \to \mathbf{Cat}_k,$$

and we can attempt to formulate my whole question as follows:

Question 2. Is $F_k \colon \mathbf{Adj}_k \to \mathbf{Cat}_k$ locally full and locally faithtful?

I should add that the story feels different if we restrict to finite-dimensional vector spaces, because then $i_V \colon V \to V^{\ast \ast}$ is an isomorphism. I don't want to restrict to finite-dimensional vector spaces.