TITLE: If adjunction arises everywhere, where is it in the fundamental theorems?
QUESTION [23 upvotes]: MacLane's slogan "adjunction arises everywhere" is widely known, and adjunction has been identified as a key concept (maybe the key concept?) in category theory, eg, in the books by Goldblatt Topoi, Awodey Category Theory, and others: 

The notion of adjoint functor applies everything that we’ve learned up
  to now to unify and subsume all the different universal mapping
  properties that we have encountered, from free groups to limits to
  exponentials. But more importantly, it also captures an important
  mathematical phenomenon that is invisible without the lens of category
  theory. Indeed, I will make the admittedly provocative claim that
  adjointness is a concept of fundamental logical and mathematical
  importance that is not captured elsewhere in mathematics. -- David Ellerman (quoting Awodey) "Adjoints and emergence: applications of a new theory
  of adjoint functors" Axiomathes 2007

If adjunction, arises everywhere shouldn't we see more examples across the spectrum of maths? 
For the most part, it seems the example of natural isomorphism that is most widely quoted is that between the category of vector spaces, and its double dual, as discussed here. 
Even a recent book like Roman's Lattices and Ordered Sets only gives 3 examples, including the one above. 
The discussion in reply to the question "A bestiary about adjunctions" asked in Math.SSE a year ago primarily revolves around algebraic structures. 
Similarly, much of the research seems to be very abstract and algebraic in nature. 
But where is application of adjunction and universal mapping property in any of the fundamental theorems or their generalizations (eg, f.t. algebra --> Bezout's theorem). Here is a short list of fundamental theorems in Wikipedia. Or perhaps it's possible in principle but it would take a lot of effort to identify the categories, functors and related constructions (transposition, unit, counit?). 
EDIT
Here by the way is a historical timeline - implemented in Mathematica - of advances in adjunctions, motivated by Qiaochu Yuan's choice of Galois theory to address this question:

REPLY [14 votes]: Paraphrasing Lawvere a bit, semantics and syntax are adjoint on the right. Let's recall some definitions. Assume two strongly inaccessible cardinals – though probably one is enough.
Let $\textbf{FinCard}$ be the category of finite cardinals ($0, 1, 2, \ldots$) and all maps between them. (So $\textbf{FinCard}$ is a small skeleton of the category of finite sets.) A Lawvere theory is a small category $\mathbb{T}$ equipped with a bijective-on-objects functor $\textbf{FinCard} \to \mathbb{T}$ that preserves all finite coproducts. The category of Lawvere theories is the full subcategory $\mathcal{T}$ of the coslice category $(\{ \textbf{FinCard} \} \downarrow \textbf{Cat})$ spanned by Lawvere theories.
An algebra of type $\mathbb{T}$ is a product-preserving functor $\mathbb{T}^\textrm{op} \to \textbf{Set}$. A homomorphism of algebras of type $\mathbb{T}$ is a natural transformation of such functors. Write $\textbf{Mod}(\mathbb{T})$ for the (concrete) category of models of $\mathbb{T}$. 
Theorem (Lawvere). Let $\textbf{CAT}$ be the category of locally small categories, and let $\mathcal{K}$ be the full subcategory of the slice category $(\textbf{CAT} \downarrow \{ \textbf{Set} \})$ spanned by those functors $U : \mathcal{C} \to \textbf{Set}$ such that the class of natural transformations $U(-)^n \Rightarrow U(-)$ is small for every natural number $n$. Then, $\textbf{Mod}(-)$ is a functor $\mathcal{T}^\textrm{op} \to \mathcal{K}$, and it has a right adjoint $\textbf{Th}(-) : \mathcal{K}^\textrm{op} \to \mathcal{T}$, which is defined as follows: for a given object $U : \mathcal{C} \to \textbf{Set}$ in $\mathcal{K}$, $\textbf{Th}(U)$ is the opposite of the full subcategory spanned by the image of the unique product-preserving functor $\textbf{FinCard}^\textrm{op} \to [\mathcal{C}, \textbf{Set}]$ given by $1 \mapsto U$. Here, by adjointness we mean there is a bijection
$$\mathcal{K}(U, \textbf{Mod}(\mathbb{T})) \cong \mathcal{T}(\mathbb{T}, \textbf{Th}(U))$$
that is natural in $U$ and $\mathbb{T}$.
In a sense, this is a categorification of the folklore Galois correspondence between truth and provability in logic. There's another result along this vein in topos theory, namely Diaconescu's theorem. It is perhaps the closest thing to a fundamental theorem in the subject, and I will now state a special case:
Theorem (Diaconescu). Let $\mathbb{A}$ be a small category with finite limits, and let $\mathcal{E}$ be a locally small and cocomplete topos. Let $\hat{\mathbb{A}}$ be the presheaf topos $[\mathbb{A}^\textrm{op}, \textbf{Set}]$, and let $y : \mathbb{A} \to \hat{\mathbb{A}}$ be the Yoneda embedding.

For each left exact functor $F : \mathbb{A} \to \mathcal{E}$, there is a unique left exact functor $F^* : \hat{\mathbb{A}} \to \mathcal{E}$ such that $F^* y \cong F$, and $F^*$ is the left adjoint to the functor $F_* : \mathcal{E} \to \hat{\mathbb{A}}$ defined by $F_* E = \mathcal{E} (F(-), E)$.
This extends to an equivalence between the category of left exact functors $\mathbb{A} \to \mathcal{E}$ and the category of left exact left adjoint functors $\hat{\mathbb{A}} \to \mathcal{E}$, and this equivalence is pseudonatural in $\mathbb{A}$ and $\mathcal{E}$. (Here $\mathbb{A}$ is regarded as an object in the category of small finitely-complete categories and left exact functors.) 

Why is this important? Because it tells us that every presheaf topos $\hat{\mathbb{A}}$ is a classifying topos for the geometric theory of left exact functors $\mathbb{A} \to \textbf{Set}$. This sounds very abstract, but in fact Lawvere already more-or-less showed that all finitary algebraic theories are of this form. To be precise, if $\mathbb{T}$ is a finitary algebraic theory and $\mathbb{A}$ is the opposite of the category of finitely-presented models of $\mathbb{T}$ in $\textbf{Set}$, then the category of left exact functors $\mathbb{A} \to \textbf{Set}$ is equivalent to the category of (not necessarily finitely-presented) $\mathbb{T}$-models in $\textbf{Set}$. We think of $\mathbb{A}$ as the syntactic category of $\mathbb{T}$: its objects can be regarded as certain well-formed formulae in the language of $\mathbb{T}$, whose interpretation in a model of $\mathbb{T}$ is given by the left exact functor $\mathbb{A} \to \textbf{Set}$ that represents the model.