$\newcommand{Po}{{\cal P}(\omega)}$
$\newcommand{lh}{\leq_{\text{hom}}}$
If $H_i = (V_i, E_i)$ are [hypergraphs](https://en.wikipedia.org/wiki/Hypergraph) for $i = 1,2$, then a map $f:V_1 \to V_2$ is said to be a *(hypergraph) homomorphism* if $f(e_1) \in E_2$ for all $e_1 \in E_1$. We say $f:V_1\to V_2$  is an *isomorphism* if $f$ is a bijection and we have $e_1\in E_1$ if and only if $f(e_1) \in E_2$.

In this question we focus on hypergraphs $H=(\omega,E)$ (having base set $\omega$  and $E\subseteq \Po$). 

For $E_1, E_2 \subseteq \Po$ we write $E_1 \lh E_2$ if there is a homomorphism $f:(\omega, E_1)\to (\omega, E_2)$. 

**Questions.** Let $E_1, E_2 \subseteq \Po$. 

1) Is there necessarily a set $E^* \in\Po$ with $E_1, E_2 \lh E^*$ and the following property?
> Whenever $E\subseteq \Po$ such that $E_1, E_2 \lh E$, then $E^* \lh E$.

2) Dually: Is there necessarily a set $E_* \in\Po$ with $E_*\lh E_1, E_2$ and the following property?
> Whenever $E\subseteq \Po$ such that $E\lh E_1, E_2$, then $E\lh E_*$.