We consider functors to be a morphism between categories, that is objects and their morphisms.
I got an intuition for functors, where objects and their morphisms are like a one dimensional line (i.e. the only thing that can be created from this is a line, through the composition of morphisms) and using functors is like going into the second dimension.
So, naturally, I first thought that natural transformation are the functors of functors, but that doesn't seem to be the case as they are the morphisms of functors (i.e. morphisms of their own right) excluding their object.
But if they only map morphisms (without their attached objects), what happens to the objects of those morphisms? So we have components of natural transformations at specific objects, but this almost seems to me like unnecessary complication.
Why is it important for natural transformation to specifically map functors, but not their belonging categories, what would change if they did?
Best Answer
A functor from $\mathcal A$ to $\mathcal B$ must take every morphism in $\mathcal A$ to a corresponding morphism in $\mathcal B$.
A natural transformation connects one particular functor $\mathcal A\to\mathcal B$ to one particular other functor $\mathcal A\to\mathcal B$. It does not need to apply to every functor in some category of functors.
Therefore it cannot be thought of as a "functor of functors" -- it does not even take functors as input.
What you can do is think of a natural tranformation as a morphism between functors. All functors $\mathcal A\to\mathcal B$ together with the natural transformations between them make up a functor category.