Natural transformation
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A natural transformation is a way of turning one functor into another functor while 'preserving' the structure of the categories. Natural transformations can be thought of a 'morphisms between functors,' and indeed they are precisely the morphisms in functor categories.
More precisely, given two categories 
and 
, and two functors 
, then a natural transformation 
is a mapping which assigns to each object 
a morphism 
in such that for every morphism 
of , we have:![\[\varphi_Y\circ F(f) = G(f)\circ \varphi_X.\]](http://latex.artofproblemsolving.com/2/4/0/240d371946d89b12baf5bf4eb56cc383a25648dd.png)
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