TITLE: A bestiary about adjunctions
QUESTION [62 upvotes]: What is your favourite adjoint? Following Mac Lane philosophy adjoints are everywhere, so I would like to draw a (possibly but unprobably) exhaustive list of adjunctions one faces in studying Mathematics. For the sake of clarity I would like you to follow a general scheme, a very naive example of which can be the following:

Functors F and G between cats C and D
Is the adjunction a (co)reflection?
Does the left adjoint admit a left adjoint on its own?
Anything you want to add

Obviously you are totally free to expand it, revert it...
I would also like to grasp something more than a mere enumeration: i.e. listing all adjunctions $\mathbf{Groups}\leftrightarrows\mathbf{Sets}$, $\mathbf{Monoids}\leftrightarrows\mathbf{Sets}$,  $\mathbf{Mod}_R\leftrightarrows\mathbf{Sets}$ is certainly a good thing, but it would be slightly better to say that all these pairs come from a "general scheme of adjunction"
$$
\text{generated object} \dashv \text{forgetful functor}
$$
which can be (if I'm not wrong) studied for a general type of algebraic structure. Hence it would be better to write some sort of "reference card" about:

The diagonal functor $\Delta_\mathbf J\colon \mathbf C\to \mathbf C^\mathbf J$ sending $C\in\text{Ob}_\mathbf C$ into the constant diagram over $C$ admits both a left and right adjoint (direct and inverse limit).
Once you fixed a set $J$, here is an adjunction between $\mathbf{Sets}/J$ and $\mathbf{Sets}^J$ defined by functors $L\colon h\in \mathbf{Sets}/J\mapsto \big(h^\leftarrow(\{j\}\big)_{j\in J}$ and $M\colon \{H_j\}_{j\in J}\mapsto \big(\coprod_{j\in J} H_j\to J\big)\in \mathbf{Sets}/J$, which turns out to be an equivalence
There exists an adjunction between $\mathrm{PSh}(X)$ and $\mathbf{Top}/X$ for any topological space $X$ ($\text{bundle of germs}\dashv\text{(pre)sheaf of sections}$), which turns out to be an equivalence if we restrict...
Given a ring $R$ the functor $R[\;\;]\colon \mathbf{Groups}\to \mathbf{Rings}$ sending a group in its group ring admits a right adjoint, namely $U\colon R\mapsto R^\times$ (units in $R$).
The inclusion functor $\mathbf{Kelley}\to\mathbf{Top}$ admits a right adjoint, the kelleyfication of a topological space
(Following Gabriel&Zisman) The inclusion functor between (small) categories $\mathbf{cat}$ and (small) groupoids $\mathbf{Gpds}$, admits both a left adjoint ($\mathbf{C}\mapsto \mathbf{C}[\text{Mor}_\mathbf{C}^{-1}]$ in the notation used for the calculus of fractions) and a right adjoint ($\mathbf{C}\mapsto \mathbf{C}^\times$, sending a category in the groupoid obtained deleting every noninvertible arrow).
... 

Feel free to say this is a silly or boring question.

REPLY [2 votes]: The inclusion functor $U \colon \mathbf{Set} \to \mathbf{Rel}$ has a right adjoint $\mathcal{P} \colon \mathbf{Rel} \to \mathbf{Set}$ which maps sets onto their power sets and relations $r \subset X \times Y$ onto
$$ \mathcal{P}(r) \colon \mathcal{P}(X) \to \mathcal{P}(Y), \qquad S \mapsto \left\{ y \in Y : \exists \, s \in S \text{ such that } sry \right\}. $$
This is essentially the direct image functor but for relations instead of just functions. Its unit $\eta_X \colon X \to \mathcal{P}(U(X))$ is the singleton function $x \mapsto \{x\}$ and its counit $\varepsilon_Y \subset U(\mathcal{P}(Y)) \times Y$ is the reverse membership relation $S \; \varepsilon_Y \; y \Longleftrightarrow y \in S$ (coincidentally both of these relations are written with an epsilon symbol). Because products in $\mathbf{Rel}$ are disjoint unions and right adjoints are continuous this is a way to show that the Cartesian product of two power sets $\mathcal{P}(X) \times \mathcal{P}(Y)$ can be canonically identified with the power set of the disjoint union $\mathcal{P}(X \sqcup Y)$.