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References to John's lectures: lecture 9.
I had some trouble making this one, repeatedly drawing things I later realized were false. The key realization for me was that everything hinges on the possible presence of "non-injective" and "non-surjective" elements in the codomain. With that understanding, I think I got the pictures correct.
Given how many mistakes I made, you may want to be extra-wary about trusting the details on this one : )
EDIT: And I did indeed have a big misunderstanding of how \(h_*\) works. I've updated with corrections and improvements.
- 2018 06 18: Fixed errors in definition of \(h_*\). Better pictures and explanations of the adjunction behavior.
- 2018 05 26: First version.