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: F(x) \ar[d]^{\eta_x} \ar[r]^{F(f)} & F(y) \ar[d]^{\eta_y} \\ G(x) \ar[r]^{G(f)}         & G(y) }$
 * 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