===== Natural transformation =====
==== Collection ====
| @#55CCEE: context     | @#55CCEE: $F,G$ in ${\bf C}\longrightarrow{\bf D}$ |
| @#FFBB00: definiendum | @#FFBB00: $\eta$ in $F\xrightarrow{\bullet}G$ |
| @#AAFFAA: inclusion   | @#AAFFAA: $\eta:{\large\prod}_{(A:\mathrm{Ob}_{\bf C})}F\,A\to G\,A$ |
| @#55EE55: postulate   | @#55EE55: @#88DDEE: $\eta\circ F(\,f)=G(\,f)\circ\eta$ |

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: [[http://en.wikipedia.org/wiki/Natural_transformation|Natural transformation]]
==== Parents ====
=== Context ===
[[Functor]]