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

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

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