A proper coloring of the vertices of a graph $G$ is seen as a homomorphism from the graph vertices to the complete graph on the number of vertices equal to the chromatic number of the graph. Similarly, the fractional coloring or $a:b$-coloring of vertices is seen as a homomorphism from the graph vertices to a Kneser graph $K(a,b)$ that is a graph whose vertices are $b$ subsets of an $a$-set with two vertices adjacent iff they are disjoint.

I wish to ask if there is a similar homomorphism characterization of the list coloring. I think this is somewhat related to the recent area of DP-coloring or correspondence coloring as in the paper. Any hints in this regard. Thanks beforehand.