Michael wrote:
> When you say projecting Tyler onto Alice, are you equating the two? ie Alice = Tyler? I think I understand what you are saying but now confused by your answer. I can see why it works but not sure what you meant by project.

No, I mean that Tyler and Alice both map to Alice. For functors from a one-object category into \$$\mathbf{Set}\$$, a natural transformation is just a certain kind of function between sets. For natural transformations \$$\gamma : H \Rightarrow H\$$, it's an endomorphism. When I say "projecting Tyler onto Alice", I mean the function \$$\\{\textrm{Bob} \mapsto \textrm{Bob}, \textrm{Alice} \mapsto \textrm{Alice}, \textrm{Tyler} \mapsto \textrm{Alice}\\}\$$.