∀ {n : ℕ} {i j : Fin (n + 2)},
  i ≤ j →
    FreeSimplexQuiver.homRel
      (CategoryTheory.CategoryStruct.comp ((CategoryTheory.Paths.of FreeSimplexQuiver).map (FreeSimplexQuiver.δ i))
        ((CategoryTheory.Paths.of FreeSimplexQuiver).map (FreeSimplexQuiver.δ j.succ)))
      (CategoryTheory.CategoryStruct.comp ((CategoryTheory.Paths.of FreeSimplexQuiver).map (FreeSimplexQuiver.δ j))
        ((CategoryTheory.Paths.of FreeSimplexQuiver).map (FreeSimplexQuiver.δ i.castSucc)))