> Why do we need to establish some relation between functors using morphisms in a category?
Are you asking why do we need natural transformations? The short answer is that category theory would be extremely boring without natural transformations: most of the subject would be gone. Most of the interesting concepts in category theory, like adjoint functors, rely on natural transformations.
> 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.
In general the concepts of 'continuously' and 'smoothly' make no sense in a category, because a category has no [topology](https://en.wikipedia.org/wiki/Topology) (which is what we need to define 'continuously') or [smooth structure](https://en.wikipedia.org/wiki/Smooth_structure) (which is what we need to define 'smoothly').
People do study topological categories and smooth categories, and I've used those in my work on mathematical physics, but we won't get to those in this course.