Let $H = (V,E)$ be a hypergraph. For $v\in V$ we set $E_v = \{e\in E: v\in E\}$. The dual of $H$ is defined by $H^* =(E, V^*)$ is, where $V^* = \{E_v:v\in V\}$.

We say hypergraphs $H_i=(V_i, E_i)$ for $i=1,2$ are isomorphic if there is a bijection $\varphi:V_1\to V_2$ such that $\varphi(e_1)\in E_2$ whenever $e_1\in E_1$, and $\varphi^{-1}(e_2)\in E_1$ whenever $e_2 \in E_2$.

Let $[\omega]^\omega$ denote the set of infinite subsets of $\omega$, and let $E_1, E_2\subseteq [\omega]^\omega$ such that $\bigcup E_i = \omega$ for $i=1,2$. Let $H_i = (\omega, E_i)$ for $i=1,2$. If $H_1^* \cong H_2^*$, does this entail $H_1\cong H_2$?

Let $H = (V,E)$ be a hypergraph. For $v\in V$ we set $E_v = \{e\in E: v\in E\}$. The dual of $H$ is defined by $H^* =(E, V^*)$ is, where $V^* = \{E_v:v\in V\}$.

We say hypergraphs $H_i=(V_i, E_i)$ for $i=1,2$ are isomorphic, in symbols $H_1\cong H_2$, if there is a bijection $\varphi:V_1\to V_2$ such that $\varphi(e_1)\in E_2$ whenever $e_1\in E_1$, and $\varphi^{-1}(e_2)\in E_1$ whenever $e_2 \in E_2$.

Let $[\omega]^\omega$ denote the set of infinite subsets of $\omega$, and let $E_1, E_2\subseteq [\omega]^\omega$ such that $\bigcup E_i = \omega$ for $i=1,2$. Let $H_i = (\omega, E_i)$ for $i=1,2$.

Question. If $H_1^* \cong H_2^*$, do we necessarily have $H_1\cong H_2$?