Here's a bit general question:

Why do we need to establish some relation between functors using morphisms in a category?

If we consider functors F and G as "images" of a category C in a category D, I suspect that there are ways to make them feel similarly, but there is no way to smoothly move F to G along morphisms in D.

If such transformations indeed exist, they feel more like discrete jumps rather than smooth sliding, but still may preserve everything we need (or even be isomorphisms). So it seems that the "continuous" maps provided by natural transformations form a subset of all possible transformations, and obviously bear some advantages compared to other types.