My question is the reciprocal of this one: Show that if U is a principal ultrafilter then the canonical inmersion is an isomorphism between $\mathcal A$ and $Ult_{U}\mathcal A$
I also assume that $M$ is infinite. By the very nature of the canonical embedding and Los theorem, we have an elementary embedding of $M$ in $M^I/U$. I wish to prove that, if $j:I \to M$ (the canonical embedding) is onto then the ultrafilter is principal.
I have already proven the case $|I| \leq |M|$. In this case I manufacture a injective function and look at the element in $M$ that goes to this element with the canonical embedding.
But I can't conclude the principalness of $U$ if the cardinality of $I$ is bigger then the cardinality of $M$.
If ZFC is consistent, it is also consistent that if $U$ is non-principal, and $M$ is infiniite, then the canonical embedding is proper.
But under suitable large cardinal axioms, there exists a set $I$, and a non-principal ultrafilter $U$ on $I$, such that $U$ is countably complete, that is, the countable intersection of sets in $U$ is in $U$. For such an ultrafilter, and countably infinite $M$, the natural embedding of $M$ in $M^I/U$ is an isomorphism. For more information, please search under $\kappa$-complete ultrafilters and measurable cardinals.