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:This equation can also be expressed by saying that the following diagram commutes:
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