Natural transformation

A natural transformation is a way of turning one functor into another functor while 'preserving' the structure of the categories. Natural transformations can be thought of a 'morphisms between functors,' and indeed they are precisely the morphisms in functor categories.

More precisely, given two categories $\mathcal{C}$ and $\mathcal{D}$, and two functors $F,G:\mathcal{C}\to \mathcal{D}$, then a natural transformation $\varphi:F\to G$ is a mapping which assigns to each object $X\in \text{Ob}(\mathcal{C})$ a morphism $\varphi_X:F(X)\to G(X)$ in $\mathcal{D}$ such that for every morphism $f:X\to Y$ of $\mathcal{C}$, we have:\[\varphi_Y\circ F(f) = G(f)\circ \varphi_X.\]This equation can also be expressed by saying that the following diagram commutes:

[asy] draw((1.5,0)--(8.5,0),EndArrow); draw((1.5,10)--(8.5,10),EndArrow); draw((0,9)--(0,1),EndArrow); draw((10,9)--(10,1),EndArrow);  label("$G(f)$",(5,-1)); label("$F(f)$",(5,11)); label("$\varphi_X$",(-1,5)); label("$\varphi_Y$",(11,5));  label("$G(X)$",(0,0)); label("$G(Y)$",(10,0)); label("$F(X)$",(0,10)); label("$F(Y)$",(10,10)); [/asy] This article is a stub. Help us out by expanding it.