Definition:Natural Transformation/Covariant Functors

Definition
Let $\mathbf C$ and $\mathbf D$ be categories. Let $F, G : \mathbf C \to \mathbf D$ be covariant 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} \ar[r]^{\map F f} & \map F y \ar[d]^{\eta_y} \\ \map G x \ar[r]^{\map G f}                & \map G y}$

Also see

 * Definition:Functor Category