∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] {Y X : C} (f : Y ≅ X) (T : CategoryTheory.Over Y),
  CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.id T.left)
      (((CategoryTheory.Over.map f.hom).comp
              { obj := fun Z => CategoryTheory.Over.mk (CategoryTheory.CategoryStruct.comp Z.hom f.inv),
                map := fun {Y_1 Z} g => CategoryTheory.Over.homMk g.left ⋯, map_id := ⋯, map_comp := ⋯ }).obj
          T).hom =
    ((CategoryTheory.Functor.id (CategoryTheory.Over Y)).obj T).hom