## Natural transformation

### Collection

context | $F,G$ in ${\bf C}\longrightarrow{\bf D}$ |

definiendum | $\eta$ in $F\xrightarrow{\bullet}G$ |

inclusion | $\eta:{\large\prod}_{(A:\mathrm{Ob}_{\bf C})}F\,A\to G\,A$ |

postulate | $\eta\circ F(\,f)=G(\,f)\circ\eta$ |

Here, in the postulate, I've left the components ($\eta_A,\eta_B$ etc.) implicit.

### Discussion

#### Idea

Natural transformation form a collection of arrows within a single category which are compatible with the (structure preserving) functors.

#### Elaboration

If one thinks about it for a minute, the data provided with a natural transformation can in fact be reformulated as just another functor, namely in ${\bf C}\times(\bullet\to\bullet)\longrightarrow{\bf D}$. This mirrors a homotopy.

#### Notation

For any $A:\mathrm{Ob}_{\bf C}$, we write $\eta_A$ for the map $F\,A\to G\,A$. This is called the *component* of the natural transformation $\eta$ at $A$.

#### Reference

Wikipedia: Natural transformation