In CWM (and other text books), natural transformations are defined between functors $ F : C \to D $ and $ G : C \to D $, that is to say, functors whose destinations are the same Category $D$.
Why do we need this limit? Can't we define the natural transformations between functors $ F : C \to D $ and $ G : C \to E $?
Without the limit of destination, I think we can define the natural transformations as the functor $\eta : D \to E $ like:
$ C $ : Subcategory of $ Sets $ so objects of $ C $ are sets, and arrows of $C$ are functions.
(But I would like to remove this assumption)
$ F : C \to D $ (functor)
$ G : C \to E $ (functor)
$ X,Y \in Obj_C $
Natural transformation is a functor $\eta : D \to E $, such that:
for any Arrow $ f : X \to Y $ in Category $C$,
$ \eta((F(f))(F(x))) = (G(f))(\eta(F(x))) $ for any $ x \in X $
To define the equation in the last line, I assumed that $ X $ has elements. Is there any way to remove this assumption?