Definition of Adjunction in Category Theory I'm trying to understand the concept of adjunction in category theory. 
This wikipedia article says that if $F$ is a functor from $\mathcal{D}$ to $\mathcal{C}$ and $G$ is a functor from $\mathcal{C}$ to $\mathcal{D}$, then we say $F$ is left-adjoint (or $G$ is right-adjoint) when:

there is a bijection between $\hom_{\mathcal{C}}(F(Y), X)$ and $\hom_{\mathcal{D}}(Y, G(X))$, which is natural for every objects $X\in\mathcal{C}$ and $Y\in\mathcal{D}$.

I'm trying to figure out what "natural" means in this context. 
I'm familiar with the notion of natural transformation $\eta$ between functors $F_1$ and $F_2$ in which for every object $F_1(A)$ there is a morphism $\eta_A:F_1(A)\to F_2(A)$ such that if $f:A\to B$, then $\eta_B\circ F_1(f)=F_2(f)\circ \eta_A$. 
But I don't know how to apply it in this context. 
Any ideas? Thanks!
 A: For completeness, the point is that you can define two functors
$$ U,V : \mathcal{D}^\text{op} \times \mathcal{C} \to \mathbf{Set} $$
by the formulas
$$ U(d,c) = \hom_\mathcal{C}(F(d), c) $$
$$ V(d,c) = \hom_\mathcal{D}(d, G(c)) $$
and these two functors should be naturally isomorphic.

If you're not familiar with the hom-functor, for any category $\mathcal{C}$, $\hom$ is a functor
$$ \mathcal{C}^\text{op} \times \mathcal{C} \to \mathbf{Set} $$
On objects $X$ and $Y$, $\hom(X,Y)$ is the set of morphisms from $X$ to $Y$.
For an object $X$ and morphism $f : Y \to Z$, you have
$$ \hom(X,f) : \hom(X,Y) \to \hom(X,Z) : g \mapsto f \circ g $$
For a morphism $h : W \to X$ and an object $Y$,
$$ \hom(h,Y) : \hom(X,Y) \to \hom(W,Y) : g \mapsto g \circ h $$
And for a pair of morphisms $h : W \to X$ and $f : Y \to Z$,
$$ \hom(h,f) : \hom(X,Y) \to \hom(W,Z) $$
is simply the composite
$$ \hom(h,f) = \hom(h,Z) \circ \hom(X, f) = \hom(W,f) \circ \hom(h,Y)$$
or more simply,
$$ g \mapsto f \circ g \circ h $$
A: I'm going to assume that $G$ is covariant, but similar statements hold regardless of what types of functors each is. If you have a morphism $X\to X'$ in $\mathcal C$, then you have a morphism $G(X)\to G(X')$ in $\cal D$, and we say it's natural in $X$ if for each $Y\in\cal D$ the following diagram commutes:
$$\begin{array}{cccc}
\hom_{\cal C}(F(Y),X) & \xrightarrow{} & \hom_{\cal D}(Y,G(X)) \\
\downarrow & & \downarrow \\
\hom_{\cal C}(F(Y),X') & \xrightarrow{} & \hom_{\cal D}(Y,G(X'))
\end{array}$$
and similarly, it's natural in $Y$ if for each morphism $Y\to Y'$ in $\cal D$ the analogous diagram commutes.
A: The most concrete definition is as an initial object in a comma category. Even more concretely phrased, it's "like the free group functor $F: \mathbf{Set} \to \mathbf{Grp}$ with the forgetful functor $U: \mathbf{Grp} \to \mathbf{Set}$".
The free group functor takes a set $X$ to its free group $FX$, and a function $f: X \to Y$ to the "do the function $f$ pointwise to each letter of the word" homomorphism $FX \to FY$.
The forgetful functor just takes a group and views its underlying set, and it takes a group homomorphism and calls it a function instead.

Let $X$ be a set and $G$ a group.
Then there's an inclusion function $\iota_X: X \to UFX$ sending $x$ to the word $(x)$ of length $1$. (Recall that $UFX$ is the set of all words on the letters in $X$.)
Then the statement that $U$ and $F$ are adjoint is the same as the statement that for every function $f: X \to UG$ there is a unique group homomorphism $g: FX \to G$ such that $f = (Ug) \circ \iota_X$.
Hopefully you can convince yourself that this fact is actually true of groups and sets by contemplating it with a cup of hot chocolate. This is how I prefer to think of the adjoint, because it's so concrete in a world where everything is alarmingly abstract.
