Can there be more than one natural transformation between two functors?

Question

Suppose $\eta : S \dot{\rightarrow} T$ is a natural transformation.

Question: Is it possible that there could be another natural transformation $\eta' : S \dot{\rightarrow} T$ s.t. $\eta \ne \eta'$?

Answer 1

Yes, in general it is possible. For example, consider two constant functors, so that $S$ sends every object to an object $s$ and $T$ send every object to an object $t$, and both of them send every morphism to the identity; then any map $s\to t$ is a natural transformation $S\dot{\to}T$.

For another example, there are (at least) two transformations from the identity functor on the category of vector spaces to itself : the zero transformation and the identity.