Natural Transformations

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Natural Transformations

Definition 3.6  

A natural transformation from $ Y:\mathscr{C}\rightarrow set$ to $ X:\mathscr{C}\rightarrow set$ is an assignment of an arrow $ N:Y\rightarrow X$ that associates to each object A in $ \mathscr{C}$ an arrow $ N_A:Y(A)\rightarrow X(A)$ in Set such that, for any $ \mathscr{C}$-arrow $ f:A\rightarrow B$ the following diagram commutes

$\displaystyle \xymatrix{


$\displaystyle N_A o Y(f)=X(f) o N_B$    

where $ N_A:Y(A)\rightarrow X(A)$ are the components on N, while N is the natural transformation.
From this diagram it is clear that the two arrows $ N_A$ and $ N_B$ turn the Y-picture of $ f:A\rightarrow B$ into the respective X-picture.

Cecilia Flori 2007-01-02