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Let $L$ be a first order language, let $M,N$ be $L$-structures such that $N$ is finite. Let $h: N \to M$ be an elementary embedding i.e. for every $L$-formula $\psi[x_1,...,x_n]$ , we have $N \vDash \psi[a_1,...,a_n] \Leftrightarrow M\vDash \psi [h(a_1),...,h(a_n)],\forall a_1,...,a_n \in N$.

Then is it true that $h: N \to M$ is an isomorphism of $L$-structures ?

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Yes. You can formalize a first-order formula which asserts a structure has $|N|$ elements. $N$ satisfies this formula, hence $M$ also does. Since $h$ is one-to-one, $h$ is bijective. Now we get $h$ is an isomorphism, by elementarility.

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