Let $G$ and $H$ be graphs. Is there a name for a function $f$ which
- Maps each vertex $x$ of $G$ to a vertex $f(x)$ of $H$
- Maps each edge $e \in E(G)$ with endpoints $x$ and $y$ to a path $f(e)$ between $f(x)$ and $f(y)$
In other words, $f$ is like a graph homomorphism, but edges can be mapped to any path with the right endpoints.