Categorical similarities between Galois theory and Hilbert's Nullstellensatz

In the following, I am going to compare two correspondences: $\textbf{correspondence between intermediate fields of $L/K$ and subgroups of ${\rm Aut}_K(L)$}$ vs $\textbf{correspondence between ideals of $R=k[x_1,\cdots,x_n]$ and algebraic subsets of $\mathbb{A}^n_k$}.$

While doing this, I will show you some similarities, which, to me, are more than a coincidence. Then, I will ask you how we are able to explain such similarities from a broader perspective--maybe a categorical explanation.

Consider the following four maps: $$\mathcal{Z}:R \to \mathbb{A}^n_k$$ sending subset $T$ of the polynomial ring over algebracially closed field to their common zeros $\mathcal{Z}(T)$, $$\mathcal{I}:\mathbb{A}^n_k \to R$$ sending subset $Y$ of the affine space with Zariski topology to the radical ideal whose zero set is $\bar{Y}$, $$\mathcal{G}:L \to {\rm Aut}_K(L)$$ sending intermediate field $E$ to $\mathcal{G}(E)={\rm Aut}_E(L)$, and $$\mathcal{F}:{\rm Aut}_K(L) \to L$$ sending subgroup $H$ to the fixed field $\mathcal{F}(H)=L^H$. Note also that we extend $\mathcal{F}$ to send a subset $S$ to the fixed field $\mathcal{F}(\langle S \rangle)$ of the smallest subgroup $\langle S\rangle$ containing $S$. Observe the following similarities:

Similarity 1:

We have \begin{align}\mathcal{ZIZ}&=\mathcal{Z}\\ \mathcal{GFG}&=\mathcal{G}.\end{align} For any ideal $a \subset R$, $\mathcal{ZIZ}(a)=\mathcal{Z}(a)$. Similarly, for any intermediate field $E$, $\mathcal{GFG}(E)=\mathcal{G}(E)={\rm Aut}_E(L)$.

Similarity 2:

We have \begin{align}\mathcal{IZI}&=\mathcal{I}\\ \mathcal{FGF}&=\mathcal{F}.\end{align} For any subset $Y \subset \mathbb{A}^n_k$, $\mathcal{IZI}(Y)=\mathcal{I}(Y)=\sqrt{\mathcal{I}(Y)}$. Similarly, for any subset $S \subset {\rm Aut}_K(L)$, $\mathcal{FGF}(S)=\mathcal{F}(S)=L^{\langle S\rangle}$.

Similarity 3:

For any subsets $Y \subset \mathbb{A}^n_k$ and $S \subset {\rm Aut}_K(L)$, we have \begin{align}\mathcal{ZI}(Y)&\supset Y\\ \mathcal{GF}(S)&\supset S.\end{align}

Similarity 4:

For any ideal $a \subset R$ and intermediate field $L/E/K$, we have \begin{align}\mathcal{IZ}(a)&\supset a\\ \mathcal{FG}(E)&\supset E.\end{align}

Similarity 5:

If we resterict ourselves to closed subsets $Y \subset \mathbb{A}^n_k$ and subgroups $ S \subset {\rm Aut}_K(L)$, then we have \begin{align}\mathcal{ZI}&=id\\ \mathcal{GF}&=id.\end{align}

Similarity 6:

If we resterict ourselves to radical ideals $a \subset R$ and Galois extensions $L/E$, then we have \begin{align}a=\sqrt{a} &\Longrightarrow &\mathcal{IZ}&=id\\ L^{{\rm Aut}_E(L)}=E &\Longrightarrow &\mathcal{GF}&=id.\end{align}

I know that Similarities 5 and 6 are just special cases of the former similarities, but still I wanted to distinguish them. It is like the subsets of $\mathbb{A}^n_k$ behave similarly to the subsets of ${\rm Aut}_K(L)$, and the algebraic subsets of $\mathbb{A}^n_k$ behave similarly to the subgroups of ${\rm Aut}_K(L)$. Furthermore, ideals behave like extensions $L/E$ and radical ideals behave like Galois extensions $L/E$. How do you explain all these similarities? To me, it's like one part of math copy-pasted with adjustments to another part. Is there a categorical explanation for this? Something like $\mathcal{Z}$ and $\mathcal{G}$ are functors with very similar features, and also $\mathcal{I}$ and $\mathcal{F}$ their inverse functors again with similar features?