Category of 'marked' vector spaces? CONTEXT
From some time on I've been loosely thinking about conjugation of complex numbers as an autofunctor of the category of $\Bbb C$-vector spaces. Although I don't remember having found it anywhere in the literature, it just seemed an obvious possibility for me, and a particularly useful one when thinking about Hermitian forms, thus sending all the information about antilinearity of certain maps (a notion that seemingly results in two types of morphisms; for instance, how would you add a linear map and an antilinear one?) to the 'inside' of some 'extra' objects, keeping every single map plainly 'linear'.
Today, for a simple reason I won't mention, I finally stumped on the need of formalizing such a way of thinking about conjugation. As I had expected, it was not very hard to do so. Though, I'm not sure if the definition I managed to produce will endure all the difficulties Mathematics shall impose to it (haha), so I decide to share it here, expecting either to find some references on the subject, or at least to get some hints on the problems I'll have to overcome. I tried to write it in a slightly more general way.
DEFINITIONS
Let $E/F$ be a field extension and let ${\rm Aut}(E/F)$ denote its automorphism group. We shall denote by $Vect_{E/F}$ the category of $E$-vector spaces 'marked' with elements of
${\rm Aut}(E/F)$. More precisely, the objects of $Vect_{E/F}$ are pairs $(V,\alpha)$, where $V$ is an $E$-vector space and $\alpha\in{\rm Aut}(E/F)$, and the morphisms in
$Vect_{E/F}\big((V_1,\alpha_1),(V_2,\alpha_2)\big)$ are functions
$\lambda:\mathcal S(V_1)\to\mathcal S(V_2)$ such that the following diagrams commute:
$$\require{AMScd}
\begin{CD}
\mathcal S(V_1)\times\mathcal S(V_1)@>(\lambda,\lambda)>>\mathcal S(V_2)\times\mathcal S(V_2)\\
@V\mathcal S(\sigma_1)VV&@VV\mathcal S(\sigma_2)V\\
\mathcal S(V_1)@>>\lambda>\mathcal S(V_2)
\end{CD}\qquad\qquad
\begin{CD}
\mathcal S(E)\times\mathcal S(V_1)@>\theta>>\mathcal S(E)\times\mathcal S(V_2)\\
@V\mathcal S(\mu_1)VV&@VV\mathcal S(\mu_2)V\\
\mathcal S(V_1)@>>\lambda>\mathcal S(V_2)
\end{CD}$$
where $\mathcal S:Vect_E\to Sets$ denotes the forgetful functor,
$\sigma_i:V_i\times V_i\to V_i$ and
$\mu_i:E\times V_i\to V_i$, $i\in\{1,2\}$ denote the usual addition of vectors and scalar multiplication of $E$-vector spaces, and
$$\theta:=(e,v_1)\mapsto\big(\mathcal S(\alpha_2\alpha_1^{-1})(e),\lambda(v_1)\big).$$
Note that this definition keeps the usual definition of $E$-vector spaces, but changes the definition of morphism of vector spaces in a way that, at least it seems to me, cannot be promptly obtained from the usual one.
Finally, let $\beta\in{\rm Aut}(E/F)$. Then we can define a functor
$\mathcal F_\beta:Vect_{E/F}\to Vect_{E/F}$ simply by putting
$$\mathcal F_\beta:(V,\alpha)\mapsto(V,\alpha\beta)\qquad\mathcal F_\beta:\lambda\mapsto\lambda.$$
QUESTIONS

*

*Have you ever found definitions similar to the above ones in the literature? Where?


*If you didn't, what problems (of any kind) involving these definitions come to your mind?


*In the case $E/F=\Bbb C/\Bbb R$, denote $V:=(V,1)$ and $\overline V:=(V,-)$. Would you agree that a nondegenerate Hermitian form on $V$ can be described as an isomorphism $\varphi:V\to\overline V^*$ such that $\overline\varphi^*=\varphi$, where $*$ denotes the dual?
 A: *

*Your definition is a special case of the Grothendieck construction of a (fibered) category out of a contravariant pseudo-functor to the category of categories.

*N/A

*No, such an isomorphism would only be a non-degenerate sesqui-linear pairing. You would need symmetry and positive-definite conditions to get a Hermitian form.

Here's some details regarding 1.
For each automorphism $\beta$ of $E$ you have a functor $F_\beta\colon\mathrm{Vect}_E\to\mathrm{Vect}_E$ sending an $E$-vector space $V$ to a new vector space $\beta^*V$ that has the same underlying set and addition of vectors, but scaling given by $(e,v)\mapsto\beta(e)v$; this is functorial because if $V\to W$ ie $E$-linear, then it is also $E$-linear as a function $\beta^*V\to\beta^*W$.
An important feature of these functors is that $F_\beta\circ F_\alpha=F_{\alpha\circ\beta}$ and that $F_{\mathrm{id}}=\mathrm{id}$. In other words, we have a contravariant functor from the automorphism group of $E$ considered as a one-object category to the category of categories, sending the elements of the automorphism group to endofunctors on the category of $E$-vector spaces.
What you are doing is then defining for each automorphism $\beta$ of $E$ a $\beta$-homomorphism of $E$-vector spaces to be an $E$-linear morphism $\lambda\colon V_1\to\beta^*V_2$. The labels here are kind of a red herring: what you call a morphism from $(V_1,\alpha_1)$ to $(V_2,\alpha_2)$ is now an $\alpha_2\circ\alpha_1^{-1}$-morphism from $(\alpha_1^{-1})^*V_1$ to $(\alpha_2^{-1})^*V_2$.
