 Kernel list-coloring lemma

In a kernel-perfect directed graph, lists of size at least $d^{+}(v)+1$ admit a proper graph coloring. Choose a color $a$ and a digraph kernel in the subgraph on vertices whose lists contain $a$. Color that kernel $a$, delete it, and remove $a$ from the remaining lists. Every vertex losing a color also loses an outgoing neighbour; hence the inequality persists. Induction on the vertex count finishes the proof.