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The two-cardinal transfer property $(\kappa,\lambda)\rightarrow(\kappa',\lambda')$ says "any structure of type $(\kappa,\lambda)$ has an elementarily equivalent structure of type $(\kappa',\lambda')$". If $\kappa\geq\kappa'$ and $\lambda\geq\lambda'$, it is natural to consider a stronger property $(\kappa,\lambda)\twoheadrightarrow(\kappa',\lambda')$ which says "any structure of type $(\kappa,\lambda)$ has an elementary substructure of type $(\kappa',\lambda')$", also known as Chang's conjecture. See e.g. Foreman's paper for what is known about these transfer properties.

My question is if we can consider the opposite direction, i.e., if $\kappa\leq\kappa'$ and $\lambda\leq\lambda'$, let $(\kappa,\lambda)\rightarrowtail(\kappa',\lambda')$ denote the statement "any structure of type $(\kappa,\lambda)$ has an elementary extension of type $(\kappa',\lambda')$". Has this property been studied? Or is it equivalent to $(\kappa,\lambda)\rightarrow(\kappa',\lambda')$?

We can further strengthen this to $(\kappa,\lambda)\rightsquigarrow(\kappa',\lambda')$ which requires the elementary embedding to be an ultrapower. Does this property reduce to the "one-cardinal" version? Namely is it really stronger than "there is an ultrapower of $\kappa$ of size $\kappa'$ and an ultrapower of $\lambda$ of size $\lambda'$, using (possibly different) ultrafilters on the same index set"?

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Suppose $\kappa\le\kappa',\lambda\le\lambda'$. By expanding the language, we can get the elementary extension version from the usual version. Specifically, let $\mathfrak{A}$ be a structure of cardinality $\kappa$, whose language includes a unary relation symbol $U$ with $U^\mathfrak{A}$ of cardinality $\lambda$. Now let $\mathfrak{A}'$ be the expansion of $\mathfrak{A}$ (so in particular, same underlying set and $U$-interpretation) gotten by naming each element $a$ with a constant symbol $c_a$. If $\mathfrak{B}\equiv\mathfrak{A}'$ then we get an elementary embedding $\mathfrak{A}'\rightarrow\mathfrak{B}:a\mapsto c_a^\mathfrak{B}$, so the non-elementary Chang principle applied to the expanded structure $\mathfrak{A}'$ yields the elementarity-demanding Chang principle applied to the original structure $\mathfrak{A}$.

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