In the proof of the following theorem:
Theorem 29 (Bukovský-Hechler): Let $\kappa, \lambda$ be infinite cardinals such that $\mathrm{cf}(\kappa) \le \lambda$ and $\mathrm{cf}(\kappa) < \kappa$. Denote $\displaystyle \sum_{\alpha < \kappa} |\alpha|^\lambda = \mu$.
(a) If there exists $\alpha_0 < \kappa$ such that $|\alpha|^\lambda = |\alpha_0|^\lambda$ for all $\alpha$ satisfying $\alpha_0 \le \alpha < \kappa$, then $\kappa^\lambda = \mu$.
(b) If for each $\alpha < \kappa$ there exists $\beta$ such that $\alpha < \beta < \kappa$ and $\alpha^\lambda < \beta^\lambda$, then $\kappa^\lambda = \mu^{\mathrm{cf}(\mu)}$.
there appears to be a typo in the proof of part (b). Can you confirm this? The proof is the following:
I don't quite believe that $H$ maps into ${^{\mathrm{cf}(\kappa)}}F$. Assume $\gamma_\xi$ is somewhere between $\alpha < \gamma_\xi < \kappa$. Then $f(\beta)$ could also be somewhere above $\alpha$ so that $g \notin F$. I am not sure what am missing but I think the definition of $H$ should depend on $\alpha$. The fix I propose is the following:
$ g(\beta) = \begin{cases} f(\beta) & \gamma_\xi < \alpha \\ 0 & \text{otherwise}\end{cases}$
Thanks for your help.