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}\\}\\).

> 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}\\}\\).