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Consider $(\kappa^{+++},\kappa^{++}) \twoheadrightarrow (\kappa^+,\kappa)$where $\kappa$ is an uncountable regular cardinal. By a result of Shelah's wehave $\operatorname{cof}(X \cap \kappa^{++}) = \kappa$ for almost all $X\subset \kappa^{+++}$ witnessing this. Here we consider the question if therecould be a similar result for $X \cap \kappa^+$. We use this discussion to givean interesting example of a pseudo Prikry forcing answering a question ofSinapova.

Some basic thoughts on the cofiality of Chang structures with an application to forcing