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Homenatural transformation
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natural transformation
Definition. Let $\mathcal{C}$ and $\mathcal{D}$ be categories, and let $S,T:\mathcal{C}\to\mathcal{D}$ be covariant functors. Then suppose that for every object $A$ in $\mathcal{C}$ one has a morphism $\eta_{A}:S(A)\to T(A)$ in $\mathcal{D}$ such that for every morphism $\alpha:A\to B$ in $\mathcal{C}$ the following
$\xymatrix@+=4pc{S(A)\ar[d]_{{S(\alpha)}}\ar[r]^{{\eta_{A}}}&T(A)\ar[d]^{{T(% \alpha)}}\\ S(B)\ar[r]^{{\eta_{{B}}}}&T(B)}$ 
is commutative. Then we variously write
$\eta:S\dot{\to}T\quad\mbox{ or }\quad\eta:S\Rightarrow T\quad\mbox{ or }\quad% \eta:S\to T$ 
and call $\eta$ a natural trasformation from $S$ to $T$.
One may think of a natural transformation $\eta:S\to T$ as a ‘function’ from the class of objects of $\mathcal{C}$ to the class of morphisms of $\mathcal{D}$.
As a first example, for every functor $S:\mathcal{C}\to\mathcal{D}$, we can associate the natural transformation $1_{S}:S\to S$ (the identity natural transformation on $S$) that assigns every object $A$ of $\mathcal{C}$, the corresponding identity morphism $1_{{S(A)}}$.
Natural transformations are composed in a similar manner to morphisms, but they are nevertheless defined as correspondences between both objects and morphisms as shown in the square commutative diagram depicted above.
More precisely, given three functors $R,S,T:\mathcal{C}\to\mathcal{D}$, and two natural transformations, $\tau:R\to S$ and $\eta:S\to T$, we define the composition of $\tau$ with $\eta$, written $\eta\bullet\tau$, as a class of morphisms in $\mathcal{D}$ given by
$(\eta\bullet\tau)_{A}:=\eta_{A}\circ\tau_{A},$ 
for every object $A$ in $\mathcal{C}$. It is easy to see that $\eta\bullet\tau$ is a natural transformation, since we may “compose” two commutative squares and obtain a third one:
$\xymatrix@+=4pc{R(A)\ar[d]_{{R(\alpha)}}\ar[r]^{{\tau_{A}}}&S(A)\ar[d]_{{S(% \alpha)}}\ar[r]^{{\eta_{A}}}&T(A)\ar[d]^{{T(\alpha)}}\ar@{}[dr]{=}&R(A)\ar[d]% _{{R(\alpha)}}\ar[r]^{{\eta_{A}\circ\tau_{A}}}&T(A)\ar[d]^{{T(\alpha)}}\\ R(B)\ar[r]^{{\tau_{{B}}}}&S(B)\ar[r]^{{\eta_{{B}}}}&T(B)&R(B)\ar[r]^{{\eta_{B}% \circ\tau_{B}}}&T(B)}$ 
It is easy to see that the composition “operation” on natural transformations is associative:
$(\zeta\bullet\eta)\bullet\tau=\zeta\bullet(\eta\bullet\tau)$ 
for natural transformations $\tau:R\to S$, $\eta:S\to T$, and $\zeta:T\to U$. In addition, any identity natural transformation acts as a compositional identity: if $\tau:R\to S$ and $\eta:S\to T$, then
$1_{S}\bullet\tau=\tau\qquad\mbox{ and }\qquad\eta\bullet 1_{S}=\eta.$ 
Remarks.

Natural transformations arise frequently in mathematics. Here’s a “concrete” example of a natural transformation: the boundary map $H_{n}(X,A)\to H_{{n1}}(A)$ in a homology theory. By definition, every morphism of presheaves is a natural transformation. More prosaically, the determinant $\det:\GL_{n}\dot{\to}(\ )^{*}$ is natural.

Natural transformations are sometimes called also functorial morphisms especially in applications related to the category theory development line pursued by Charles Ehresmann and the ‘Nicolas Bourbaki’ group; this is also a natural translation of the same concept from French, viz. (ref. [3]).
References
 1 A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.
 2 S. Mac Lane, Categories for the Working Mathematician (2nd edition), SpringerVerlag, 1997.
 3 C. Ehresmann, Catégories et Structures. Dunod: Paris , 1965.
 4 C. Ehresmann, Catégories doubles des quintettes: applications covariantes , C.R.A.S. Paris, 256: 18911894, 1963.
 5 S. Eilenberg and S. Mac Lane, The General Theory of Natural Equivalences, Transactions of the American Mathematical Society 58: 231294, 1945.
 6 B. Mitchell., Theory of Categories, Academic Press: New York and London.
Mathematics Subject Classification
18A25 no label found1800 no label found18A05 no label found Forums
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