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!