TITLE: Philosophy or meaning of adjoint functors
QUESTION [15 upvotes]: I have a question about the definition of the (left) adjoint of a functor.I am trying to understand the philosophy and reason of the definition of adjoint functors.
If I understand correctly the situation is as follows: Suppose one has the category $\mathcal{C}$ with an object $A$ and a category $\mathcal{D}$ with an object $B$ and functors $F: \mathcal{C} \to \mathcal{D}$ and a functor $G: \mathcal{D} \to \mathcal{C}$. 
Now one can wonder how to make kind of cross morphism from $A$ to $B$. The two natural ways would be to push $A$ to $\mathcal{D}$ using $F$ and take a morphism $\phi$ there. But there is another equally natural way to this. Push $B$ to $\mathcal{C}$ by $G$ and take a morphism $\psi$ between $A$ and $GB$. 
The natural equivalence then translates approach one in to approach two and back by making a correspondence between $\phi$ and $\psi$. 
To make it a bit sensible one demands that the functors $F$ and $G$ are not arbitrary, but somehow linked together in the sense that linking $A_1 \to A$ with $B \to B_1$ is done on essentially( up to correspendance by a unit) the same way by pushing $A$ to $\mathcal{D}$ or pushing $B$ to $\mathcal{C}$. 
Is this indeed the philosophy behind adjoint functors?

REPLY [10 votes]: One should also mention the utility of adjoint functors for explicit computation, which happens because left adjoints preserve colimits and right adjoints preserve limits. 
Consider for example the forgetful functor $Ob$ from groupoids to sets. This has a right adjoint, which to a set $X$ assigns what is called sometimes the indiscrete or coarse groupoid, say  $Ind(X)$,  on $X$, which has exactly one morphism between any two objects. 
Now suppose we want to compute $L=$colim$G_i$ of some diagram of groupoids. Because of the above, we know that $Ob(L)=$colim$Ob(G_i)$. That is a start.