## 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

$\displaystyle \xymatrix{ A\ar[dd]^f&&Y(B)\ar[rr]^{N_B}\ar[dd]_{Y(f)}&&X(B)\ar[dd]^{X(f)}\\ &&&&\\ B&&Y(A)\ar[rr]^{N_A}&&X(A)\\ }$

i.e.

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

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