自然変換の定義

Let F,G:\mathfrak{C}\to\mathfrak{D} be functors between categories C and D.

A natural transformation \Phi from F to G consists of a family \Phi_C:F(\mathfrak{C})\to G(\mathfrak{C}) of morphisms in \mathfrak{D} which are indexed by the objects C of \mathfrak{C} so that, for each morphism f:C\to D between objects in \mathfrak{C}, the equality

holds. The elements \Phi_C are called the components of the natural transformation.

If all the components \Phi_C are isomorphisms in \mathfrak{D}, then \Phi is called a natural isomorphism between F and G. In this case, one writes \Phi:F \simeq G.

さっぱりわからん