∀ (Y : TopCat),
  CategoryTheory.CategoryStruct.comp
      ({ app := fun x => id, naturality := TopCat.adj₁._proof_1 }.app ((CategoryTheory.forget TopCat).obj Y))
      ((CategoryTheory.forget TopCat).map
        ({ app := fun X => TopCat.ofHom { toFun := id, continuous_toFun := ⋯ }, naturality := TopCat.adj₁._proof_2 }.app
          Y)) =
    CategoryTheory.CategoryStruct.id ((CategoryTheory.forget TopCat).obj Y)