@Keith – my thinking is that:

— a _functor_ **1** \$$\rightarrow\$$ **C** is simply an _object_ of **C**

— so a _natural transformation_ between such functors is simply a _map_ between such objects

The most useful intuition here, it seems to me, is that a functor from **B** to **C** is a "picture of **B** in **C**".

And a natural transformation of such functors is a way of "sliding" the first picture onto the second along judiciously chosen **C**–morphisms.

In the case where **B** = **1** the picture is just a single object, and the slide is just a single map in **C**.

In the case where **B** = **0** the picture is empty. There is always exactly one such empty picture for every category **C**.

In the case where **C** = **1** there is exactly one possible picture (collapse everything down to the point).

In the case where **C** = **0** there is no "canvas" to draw the picture upon, so there is no picture/functor – unless **B** is also **0**.