A concrete intuition for Galois Connection 
A Galosis connection between preorders $P$ and $Q$ is a pair of monotone maps $f: P \to Q$ and $g: Q \to P$ such that $f(p) \leq q$ iff $ p \leq g(q)$. We say that $f$ is the left adjoint and $g$ is the right adjoint of the Galosis connection.
Page-27, An Invitation to applied Category theory

I am trying to find a concrete example of what type of idea the above definition is to capture. Could an example in Real life be given where can observe a Galosis connection? I am looking for something along the lines of the example given for monotone maps in the book:


Page-19

 A: Say you and some friends are trying to choose a restaurant to eat at. You might build a table whose rows are restaurants, and whose columns are food items that you and your friends are interested in:
$$
\begin{array}{c|cccc}
                            & \text{Burgers} & \text{Egg Rolls} & \text{Fried Rice} & \text{Dessert} \\ \hline
\text{Some Pub}             & ✔              & ✔                &                   & ✔       \\
\text{Chinese Takeout}      &                & ✔                & ✔                 &         \\
\text{Cheesecake Warehouse} & ✔              & ✔                & ✔                 & ✔      
\end{array}
$$
This gives rise to a natural galois correspondence between the powerset lattices
$$
\{ \text{subsets of restaurants} \} \leftrightarrow \{ \text{subsets of food choices} \}
$$
Explicitly, to any $R \subseteq \{ \text{restaurants} \}$, write $f(R) = \{ \text{food available at some restaurant in $R$} \}$.
And to any $F \subseteq \{ \text{food choices} \}$, we have $g(F) = \{ \text{restaurants whose menu lies entirely in $F$} \}$
Now, I'll leave it as a quick exercise to check that $(f,g)$ really is a galois pair. That is, $f(R) \leq F$ if and only if $R \leq g(F)$.
The point, though, is that this lets us convert questions about restaurants to questions about food, and vice versa. For instance, if $P = \{ \text{peanut-free food} \}$, then someone with allergies might want to know which restaurants have entirely peanut-free menus. That is, they might want be interested in $g(P)$, and this galois correspondence can help them study it.
Obviously there's nothing special here about food. Given any relation $R \subseteq A \times B$, where we think of $(a,b) \in R$ as saying that "$a$ and $b$ are related" (here, a restaurant was related to a food item exactly when it serves that item), we can get a galois correspondence between $\mathcal{P}(A)$ and $\mathcal{P}(B)$. Our relation here was fairly simple, but you can imagine for more complicated relations, this adjunction says more.
For instance, if you know a bit of group theory, you might study the relation between groups and "sentences in the langauge of groups" where $G$ is related to $\varphi$ exactly when $\varphi$ is true of $G$ (written $G \models \varphi$). In this case, the adjunction described above relates sentences consistent in a family of groups with groups whose theory is contained entirely in some set of sentences.
For more information (about galois connections in general, not the model theoretic example I just gave), you should look at chapters $3$ and $7$ in Davey and Priestley's Introduction to Lattices and Order.

I hope this helps ^_^
