  Given a countable amenable group $G$, a F\o lner sequence $(F_N) \subseteq
G$, and a set $E \subseteq G$ with $\bar{d}_{(F_N)}(E)=\limsup_{N \to \infty}
\frac{|E \cap F_N|}{|F_N|}>0$, Furstenberg's correspondence principle
associates with the pair $(E,(F_N))$ a measure preserving system
$(X,\mathcal{B},\mu,(T_g)_{g \in G})$ and a set $A \in \mathcal{B}$ with
$\mu(A)=\bar{d}_{(F_N)}(E)$, in such a way that for all $r \in \mathbb{N}$ and
all $g_1,\dots,g_r \in G$ one has $\bar{d}_{(F_N)}(g_1^{-1}E \cap \dots \cap
g_r^{-1}E)\geq\mu((T_{g_1})^{-1}A \cap \dots \cap (T_{g_r})^{-1}A)$. We show
that under some natural assumptions, the system $(X,\mathcal{B},\mu,(T_g)_{g
\in G})$ is unique up to a measurable isomorphism. We also establish variants
of this uniqueness result for non-countable discrete amenable semigroups as
well as for a generalized correspondence principle which deals with a finite
family of bounded functions $f_1,\dots,f_{\ell}: G \rightarrow \mathbb{C}$.
