\begin{definition}[Definition:Natural Transformation/Contravariant Functors]
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:
:$(1): \quad$ For all $x \in \mathbf C$, $\eta_x$ is a morphism from $\map F x$ to $\map G x$.
:$(2): \quad$ For all $x, y \in C$ and morphism $f: x \to y$, the following diagram commutes:
::$\xymatrix{
\map F x \ar[d]^{\eta_x} & \map F y \ar[d]^{\eta_y} \ar[l]^{\map F f}  \\
\map G x                 & \map G y \ar[l]^{\map G f} }$
\end{definition}