Vertical and Horizontal compositions and endofunctors

Question

Consider natural transformations between endofunctors. Two types of compositions are possible for natural transformations: vertical and horizontal. Can I say that they are the same (if the natural transformations are between endofunctors)?

Answer 1

No. In fact not only will they not generally be the same, one can make sense when the other doesn't.

For example, in a category with binary products there's a twist natural transformation $\tau:(A\times -)\to (-\times A).$ Note that vertically composing this with itself doesn't even make sense (the domain and codomain are different), but horizontal composition is fine; $(\tau*\tau)_B$ is the map $(a,(a',b))\mapsto ((b,a'),a)$.