$eq:complement$. A map $V(F) → V(X)$ is a homomorphism into the full complement exactly when it avoids, for every edge of $F$, the event that its endpoints are sent to an adjacent pair; inclusion–exclusion over those events counts the maps avoiding all of them, and the maps satisfying the events of a set $s$ of edges are the homomorphisms out of the spanning subgraph with edge set $s$.