TITLE: Why are adjoint functors common?
QUESTION [11 upvotes]: There's a famous slogan by Saunders Mac Lane in his Categories for the Working Mathematician: "Adjoints are everywhere." Over time I experienced that this is indeed true, to the point that whenever I construct two functors in the opposite direction, I immediately check if they are adjoint.
What surprises me is how often this is the case. The property of being adjoint is very special, so 'statistically' speaking, one would expect them to be rare. Why are adjoint functors so common? I'm looking for either a heuristic or a more rigorous argument.

REPLY [3 votes]: Adjoint functors are a generalisation/relaxation of the notion of an inverse. Philosophically, one could perhaps argue that many notions in mathematics are "invertible", in that we can go back to the category we started in, in a nice enough way. 
For example, when we abelianise a group $H \mapsto H^{\mathrm{ab}} = H/[H, H]$ using the abelianisation functor $\mathrm{ab} : \mathsf{Grp} \rightarrow \mathsf{Ab} $ we find that there exists an inclusion functor $G : \mathsf{Ab} \rightarrow \mathsf{Grp} $ in the reverse direction. 
Here we have an adjunction $\mathrm{ab} \dashv G$. These two functors satisfy the unit-counit conditions in the definition of adjunction; that there are natural transformations $\epsilon : \mathrm{ab} \circ G \Rightarrow 1_{\mathsf{Grp}}$ and $ \eta : 1_{\mathsf{Ab}} \Rightarrow G \circ \mathrm{ab} $ such that $ \mathrm{ab} \Rightarrow \mathrm{ab} \circ G \circ \mathrm{ab} \Rightarrow \mathrm{ab}$ and $G \Rightarrow G \circ \mathrm{ab} \circ G \Rightarrow G$ are equivalent to identity natural transformations. Of course, one can also equivalently formulate this as the natural isomorphism of hom sets
$$ \mathrm{Hom}_{\mathsf{Ab}}(\mathrm{ab}(H_1), H_2) \cong \mathrm{Hom}_{\mathsf{Grp}}(H_1, G(H_2)).   $$
I think that it's reasonable to expect a functor in the reverse direction which is compatible with the initial functor in some way; it would be too difficult to recover the group we had before abelianisation (since ultimately information is being forgotten) but the group in $\mathsf{Grp}$ that is recovered is at least related to the original group by morphisms going into it.
More precisely, let $H_1 \in \mathsf{Grp}$ be a group and abelianise to get $\mathrm{ab}(H_1) \in \mathsf{Ab}$. What is $G \circ \mathrm{ab} (H_1) \in \mathsf{Grp}$? Well, by what we've said above, 
$$  \mathrm{Hom}_{\mathsf{Grp}}(H_1, G \circ \mathrm{ab} (H_1)) \cong \mathrm{Hom}_{\mathsf{Ab}}(\mathrm{ab}(H_1), \mathrm{ab}(H_1)). $$
So the morphisms into $G \circ \mathrm{ab} (H_1)$ from what we started with, $H_1$, are "encoded" by endomorphisms between the abelianised version of what we started with $\mathrm{ab}(H_1) \rightarrow \mathrm{ab}(H_1)$.