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