Definition:Natural Transformation/Contravariant Functors

Definition
Let $\mathbf C$ and $\mathbf D$ be categories. Let $F, G : \mathbf C \to \mathbf D$ be contravariant functors.

A natural transformation $\eta$ from $F$ to $G$ is a mapping on $\mathbf C$ such that: F(x) \ar[d]^{\eta_x} & F(y) \ar[d]^{\eta_y} \ar[l]^{F(f)} \\ G(x)          & G(y) \ar[l]^{G(f)} }$
 * For all $x\in \mathbf C$, $\eta_x$ is a morphism from $F(x)$ to $G(x)$.
 * For all $x,y\in C$ and morphism $f : x \to y$, the following diagram commutes:
 * $\xymatrix{

Also see

 * Definition:Functor Category