natural transformation

{{category theory}} Given categories ''C'' and ''D'' and functors ''F'' and ''G'' both going from ''C'' to ''D'', then a transformation ''&eta;'' from ''F'' to ''G'' consists of components <br>''&eta;''_X:''F''(''X'') &rarr; ''G''(''X''), indexed by objects ''X'' of ''C'', which are morphisms in ''D''. The transformation is said to be natural if for any arrow ''f'':''X'' &rarr; ''Y'' in ''C'', the following naturality condition is satisfied: <math> \eta_Y \circ F(f) = G(f) \circ \eta_X</math>.