CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Functor.whiskerRight
      { app := fun x => CategoryTheory.Paths.of ↑((CategoryTheory.Functor.id CategoryTheory.Quiv).obj x),
        naturality := CategoryTheory.Quiv.adj._proof_1 }
      CategoryTheory.Cat.free)
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Cat.free.associator CategoryTheory.Quiv.forget CategoryTheory.Cat.free).hom
      (CategoryTheory.Cat.free.whiskerLeft
        { app := fun C => (CategoryTheory.pathComposition ↑C).toCatHom, naturality := ⋯ })) =
  CategoryTheory.NatTrans.id ((CategoryTheory.Functor.id CategoryTheory.Quiv).comp CategoryTheory.Cat.free)