TITLE: Definition of Adjunction in Category Theory
QUESTION [5 upvotes]: 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!

REPLY [7 votes]: 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,
$$ g \mapsto f \circ g \circ h $$