Natural transformation

From AoPSWiki

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: