Natural Transformation

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e. the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most basic notions of category theory and consequently appear in the majority of its applications. If F and G are functors between the categories C and D, then a natural transformation ¿ from F to G associates to every object X in C a morphism ¿X : F(X) ¿ G(X) in D called the component of ¿ at X, such that for every morphism f : X ¿ Y in C we have: eta_Y circ F(f) = G(f) circ eta_X