I proved:

Theorem $T$ is a left adjoint of both $F_{\star}$ and $F^{\star}$, with bijection which preserves the “function” part of the morphism.

The details and the proof is available in the draft of second volume of my online book.

The proof is not yet enough checked for errors.

1. porton / May 26 2018 19:05

I’ve found an error in my proof.

• porton / May 26 2018 19:45

The error was corrected.

• porton / May 26 2018 20:12

Well, one more error in the same proof 😦

• porton / May 26 2018 20:29

I’ve proposed a counterexample against the theorem (see the above link to my current draft).

2. porton / May 26 2018 20:46

The theorem was erroneous. Now the draft contains a counterexample instead of the theorem.

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