I proved:

**Theorem** $latex T$ is a left adjoint of both $latex F_{\star}$ and $latex F^{\star}$, 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.

I’ve found an error in my proof.

The error was corrected.

Well, one more error in the same proof 🙁

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

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