Why are adjoint functors common?

Theorem: Let $F : C \to D$ be a functor between locally presentable categories. Then,

• $F$ has a right adjoint if and only if it preserves small colimits
• $F$ has a left adjoint if and only if it preserves small limits and is an accessible functor

This is a rather broad class of examples for which existence of adjoints is automatic.

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 $\mathrm{ab}(H_1) \rightarrow \mathrm{ab}(H_1)$.