According to wikipedia,
...the Löwenheim–Skolem Theorem states that for every signature $σ$, every infinite $σ$-structure $M$ and every infinite cardinal number $κ ≥ |σ|$, there is a $σ$-structure $N$ such that $|N| = κ$ and
- if $κ < |M|$ then $N$ is an elementary substructure of $M$;
- if $κ > |M|$ then $N$ is an elementary extension of $M$.
Now let $\kappa$ be inaccessible. Then $V_\kappa$ agree with the ambient universe about $\aleph_1,$ furthermore $V_\kappa$ knows that $\aleph_1$ is uncountable. So by Lowenheim-Skolem, we can find a countable elementary substructure $M$ of $V_\kappa$. Since $V_\kappa$ is well-founded, so too is $M$. Hence, we can collapse $M$ to obtain an (isomorphic) transitive model $T$. But since $T$ is countable, hence $\aleph_1^T$ does not equal $\aleph_1.$ Thus certainly, it is not the case that the inclusion $T \hookrightarrow V_\kappa$ is an elementary embedding. Yet there exists an elementary embedding $T \hookrightarrow V$.
This seems kind weird. Can anyone explain what's going on here?