[kmill]

Are you familiar with how the word "vector" changes meaning depending on which vector space your talking about? The same goes for "morphism," which depends on the category. In your example of "tmap : a -> f a", the only way this would make sense is as a morphism from object 'a' of the first category to an object 'f a' of the second, but this is a semantic error (a categorical error, if you will).

It is true that functors are morphisms between categories, but that's a morphism in CAT.

> I'm confused about this.

That's how these things go :-) Anyway, natural transformations are between two functors C -> D. This is an example of a 2-morphism in a 2-category. It's hard to come up with something to say other than they're just different, but think about this: a natural transformation is a consistent choice of morphisms F X -> G X (in D) for each object X (in C), but a functor needs to know where the morphisms of C go, too.

(An intuition is a natural transformations is kind of like a continuous path transforming one functor into another. This might just make things confusing, though.)