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: