Natural Transformation

category-theory

Definition

Natural Transformation

Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $F,G:\mathcal{C}\to \mathcal{D}$ be two functors. A natural transformation $\eta :F\Rightarrow G$ assigns to every object $X\in \mathcal{C}$ a morphism

$${\eta }_X:F(X)\to G(X)$$

in $\mathcal{D}$ such that for every morphism $f:X\to Y$ in $\mathcal{C}$ the following naturality square commutes:

$$\begin{array}{ccc}F(X)& \stackrel{{\eta }_X}{\to }& G(X)\\ F(f)\downarrow & & \downarrow G(f)\\ F(Y)& \stackrel{{\eta }_Y}{\to }& G(Y)\end{array}$$

That is, $G(f)\circ {\eta }_X={\eta }_Y\circ F(f)$.

Naturality

The equation $G(f)\circ {\eta }_X={\eta }_Y\circ F(f)$ is the naturality condition. It states that moving from $F(X)$ to $G(Y)$ along either path of the square yields the same result. In other words, the transformation $\eta$ is natural precisely when it is compatible with the action of both functors on morphisms.