For (finite or infinite) undirected, simple graphs $G, H$, let

$V_{\text{Hom}} = \{f:G\to H:f\text{ is a graph homomorphism}\}$, and $E_{\text{Hom}} =\big\{\{f,g\}\subseteq V_{\text{Hom}}: \{f(v),g(v)\} \in E(H) \text{ for all } v\in V(G)\big\}.$

We set $\text{Hom}(G,H) = (V_{\text{Hom}}, E_{\text{Hom}})$.

Given any graph $G$, are there always graphs $H_1, H_2$ with more than $1$ point each such that $G\cong \text{Hom}(H_1, H_2)$?

**EDIT.** Thanks to Vidit for spotting the $1$-point solution solving the problem trivially, so I have excluded this.