TITLE: Why should I care about adjoint functors
QUESTION [72 upvotes]: I am comfortable with the definition of adjoint functors. I have done a few exercises proving that certain pairs of functors are adjoint (tensor and hom, sheafification and forgetful, direct image and inverse image of sheaves, spec and global sections ect) but I am missing the bigger picture.
Why should I care if a functor has a left adjoint? What does it tell me about the functor?

REPLY [7 votes]: It has been about half a year since I asked this question and I have learnt a lot more category theory since then. All of the above answers are excellent. One thing which everyone said is that a left (resp. right) adjoint commutes with colimits (resp. limits). Indeed, this is one of the best things about knowing that two functors are adjoints.
We can say more about the relationship between adjoints and (co)limits.

${\rm \bf Theorem:}$ Let $\mathcal{A}$ and $\mathcal{B}$ be categories and assume that $G : \mathcal{A} \to \mathcal{B}$ and $F : \mathcal{B} \to \mathcal{A}$ are functors. The following are equivalent:

There exists a natural isomorphism $\tau : {\rm Mor}(F-,-) \to {\rm Mor}(-,G-)$
There exists natural transformations $\eta : 1_{\mathcal{B}} \to G \circ F$ and $\epsilon : F \circ G \to 1_{\mathcal{A}}$ such that $$ F(B) \stackrel{F(\eta_B)}{\to} F \circ G \circ F(B) \stackrel{\epsilon_{F(B)}}{\to} F(B) $$ and $$ G(A) \stackrel{\eta_{G(B)}}{\to} G \circ F \circ G(A) \stackrel{G(\epsilon_A)}{\to} G(A) $$ are the identity morphisms
there exists a natural transformation $\eta : 1_{\mathcal{B}} \to G \circ F$ such that for all $B \in \mathcal{B}$, $\eta_B$ is a $G$-universal morphism
there exists a natural transformation $\epsilon : F \circ G \to 1_{\mathcal{A}}$ such that for all $A \in \mathcal{A}$, $\epsilon_A$ is an $F$-couniversal morphism


If any of the conditions in the above theorem are satisfied we say that $(F,G)$ is an adjoint pair. Vladimir Sotirov discussed this theorem in his answer.


$\eta_B : B \to G \circ F(B)$ is a $G$-universal morphism means that for all morphisms $f : B \to G (A)$ in $\mathcal{B}$, there exists a unique morphism $\bar{f} : F(B) \to A$ such that $$G(\bar{f}) \circ \eta_B = f$$
$\epsilon_A : F \circ G(A) \to A$ is an $F$-universal morphism means that for all morphisms $f : F(B) \to A$ in $\mathcal{A}$, there exists a unique morphism $\bar{f} : B \to G(A)$ such that
$$ \epsilon_A \circ F(\bar{F}) = f$$


Now let $I$ be a small category and $\mathcal{A}^I$ be the category whose objects are functors $I \to \mathcal{A}$ and whose morphisms are natural transformations. There is a natural functor $\Delta : \mathcal{A} \to \mathcal{A}^I$ defined as follows:


$\Delta(A) : I \to \mathcal{A}$ maps each object of $I$ to $A$ and each morphism in $I$ to the identity on $A$
if $f : A \to B$ is a morphism in $\mathcal{A}$ then $\Delta(f)$ is the natural transformation defined by $\Delta(f)_i = f$


By the above theorem, saying that every functor $D : I \to \mathcal{A}$ has a colimit is exactly the same as saying that ${\rm colim} : \mathcal{A}^I \to \mathcal{A}$ is a left adjoint of $\Delta$.  Saying that every functor $D : I \to \mathcal{A}$ has a limit is exactly the same as saying that ${\rm lim} : \mathcal{A}^I \to \mathcal{A}$ is a right adjoint of $\Delta$