Philosophy or meaning of adjoint functors

As the link of Zhen Lin says adjoint functors deal with universals.

Usually I think to adjointness situations in two different ways: as conditions of global existence of universals (related to the comment of Zhen Lin above) and as a conditions of global representability. (By Yoneda lemma these things are essentially the same but they are different, from a psychological point of view).

Let's see the first: the global existence of universals.

The following fact holds: a functor $\mathcal G \colon \mathbf A \to \mathbf X$ has a left adjoint if and only if for each $x \in \mathbf X$ exists a morphisms $\eta_x \colon x \to \mathcal G(\mathcal F(x))$, where $\mathcal F(x) \in \mathbf A$, which is universal from $x$ to $\mathcal G$ [i.e. for every other object $a \in \mathbf A$ and morphism $\tau \colon x \to \mathcal G(a)$ exists a unique $h \colon \mathcal F(x) \to a$ such that $\mathcal \tau=G(h)\circ \eta_x$].

In this case it can be show that the induced function $\mathcal F$ from the set of objects of $\mathbf X$ to the the set of objects of $\mathbf A$ can be extended to a functor $\mathcal F \colon \mathbf X \to \mathbf A$ (unique up to natural isomorphism). This condition of adjointness, the existence of one universal morphism from every object in $\mathbf A$, justifies the phrase global existance of universals.

There's also a dual result that says that a functor $\mathcal F \colon \mathbf X \to \mathbf A$ has a right adjoint $\mathcal G \colon \mathbf A \to \mathbf X$ if and only if exists a family of morphisms $\epsilon_a \colon \mathcal F(\mathcal G(a)) \to a$ each one being universal from $\mathcal F$ to $a \in \mathbf A$, i.e. for each other morphisms $\sigma \colon \mathcal F(x) \to a$ exists a unique $k \colon x \to \mathcal G(a)$ such that $\sigma = \epsilon_a \circ \mathcal F(k)$.

These facts can be rephrased in term of representable functors. So the existance of the family of universal morphisms $\langle \eta_x \colon x \to \mathcal G(\mathcal F(x))\rangle_{x \in \mathbf X}$ is equivalent via Yoneda's lemma to the representability, meaning that the functor $\mathcal G$ has a left adjoint if each functor $\mathbf X(x,\mathcal G(-)) \colon \mathbf A \to \mathbf {Set}$ is representable [that's because each universal morphism $\eta_x \colon x \to \mathcal G(\mathcal F(x))$ gives a natural isomorphism $\mathbf X(x,\mathcal G(-)) \cong \mathbf A(\mathcal F(x),-)$]. Similarly the existance of the family of universal morphisms $\langle \epsilon_a \colon \mathcal F(\mathcal G(a)) \to a\rangle_{a \in \mathbf A}$ implies that each of the functors $\mathbf A(\mathcal F(-),a)$ is representable [because the universal morphism $\epsilon_a \colon \mathcal F(\mathcal G(a)) \to a$ give a natural isomorphism $\mathbf A(\mathcal F(-),a) \cong \mathbf X(-,\mathcal G(a))$].

Representability says that each fact about morphisms of type $x \to \mathcal G(a)$ is equivalent to a fact about morphisms of type $\mathcal F(x) \to a$. Simplifying this means that we can traduce facts/problems in the category $\mathbf X$ in other facts/problems in the category $\mathbf A$, and the opposite. This enables us to transport problems in the most convenient setting.

I hope this answer was useful.

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.