Let $L$ be a first-order language consisting of only relation symbols (no function symbols, no constant symbols). Let $M = (S,\iota)$ and $M' = (S',\iota')$ be two $L$-structures. The two structures $M$ and $M'$ are said to be isomorphic if there is a bijection $h\colon S\to S'$ such that for every relation symbol $R$ in $L$ and tuple $s_1, s_2,\dots,s_{\text{arity}(R)}$ of elements in S, we have $(s_1,s_2,\dots,s_{\text{arity}(R)})\in \iota(R)$ iff $(h(s_1),h(s_2),\dots,h(s_{\text{arity}(R)})) \in \iota'(R)$. For any given finite $L$-structure $M$, show that there exists a first-order sentence $\phi_M$ such that any structure $M'$ satisfies $\phi_M$ iff $M$ and $M'$ are isomorphic.
I was trying to write a first order sentence for $\phi_M$. I don't know how to show the bijection in between $S$ and $S'$ because in $\phi_M$ the universe is $S$.
