Thank you! I knew that fact for generic ultrafilters on Boolean algebras, but I did not relate it with this context.

Thank you for your answer, but I am afraid the argument does not work. $\gamma$ is the limit of the critical sequence $\kappa_n$. Consider $A:V_\gamma\longrightarrow V_\gamma$ a function such that $A(\kappa_n)=\kappa_{n+1}$.Then $V_{\gamma+1}\vDash \forall x\in V_\gamma\exists y\in V_\gamma\ (x,y)\in A$, but $V_{\gamma}\vDash \forall x\in V_\alpha\exists y\in V_\alpha\ (x,y)\in A\cap V_\alpha$ does not hold for any $\alpha<\gamma$.

Thank you! I insist that the first paragraph does not answer the question, by the same reason that in Asaf's case, but the second does! In $L[G]$ $0^\sharp$ does not exist because $\omega_2^L=\omega_1$, but $\kappa(\omega_1^L)$ exists.

Thank you, but that does not answer the question. A negative answer would require a model in which $0^\sharp$ does not exist and there exists $\kappa(\lambda)$ for $\omega_1^L\leq \lambda<\omega_1$. $L$ does not satisfy the latter.

Maybe there is a gap! ;) Seriously, I am not sure about the significance of the similitude between Kanamori's and Jech's proof (Kunen's one and Silver's one in fact), but the fact is that Jech uses new models $M_{\alpha\beta}$ to define the embeddings $i_{\alpha\beta}$ that Kanamori introduces just as compositions $e_\beta^{-1}\circ e_\alpha$. Maybe that step is less trivial than it could seem. By the way, I am afraid that those comments are in the wrong thread!

Oh, you are very kind. I would not want to waste your time. I have found a simpler proof in Boos' paper, so understanding Kanamori's proof is not too important for me anymore, but, of course, it is a pity having understood a long proof but a single step, so I am still curious about it. But I am afraid that my curiosity is not worth wasting your time.

Congratulations!!! I was confident that you would find the right proof. I am a bit bussy now, but I will continue with your paper soon.

I have found a very simple proof of the result in W. Boos: Lectures on large cardinal axioms, 25-88 in Lecture Notes in Math. #99, Springer 1975! The main idea in the limit case is that there exist only one normal $L$-ultrafilter on a given cardinal $\kappa$.

Thank you! I did not know this result. I have edited my question after having read it.