TITLE: What are some beautiful examples of adjunctions?
QUESTION [14 upvotes]: Lately I've been very interested in finding examples of adjunctions. In particular, examples that are elementary enough for an undergrad like me to grasp. So I was wondering if perhaps you could share some of your favourite examples of adjunctions and maybe little hints and suggestions on how to explore them and their implications.  
Although I'd be very interested in examples in real analysis, number theory, and topology (something that motivates the study of locales would be very nice), I don't wish to be too restrictive, so feel free. And don't pay much attention to "for an undergrad to grasp", since being unable to understand something is a temporary thing, I hope...

REPLY [2 votes]: Here is a logic example, as a special case of Galois connections, from Lambek's "The Influence of Heraclitus on Modern Mathematics" (Scientific Philosophy Today, pp. 111-121). Lambek reports that it was first studied by Lawvere:
Let $\mathcal{P}$ be the category with formulas of propositional calculus as objects and "entailments" $\vdash$ as arrows (so that $\mathcal{P}$ is a preorder). Define $\forall p\in\operatorname{Obj}(\mathcal{P})$ endofunctors $F_p,G_p:\mathcal{P}\to\mathcal{P}$ by $F_p:q\mapsto (p \wedge q)$ and $G_p:q\mapsto (p\Rightarrow q)$ (the arrow functions are the expected ones). Then $F_p$ is a left adjoint to $G_p$, and the counit of the adjunction is the good old "modus ponens" (naturality in $p$ can be established by the so-called "parameter theorem").