/-- This is a constructor of `TopCat.GlueData` whose arguments are in terms of elements and
intersections rather than subobjects and pullbacks. Please refer to `TopCat.GlueData.MkCore` for
details. -/
def mk' (h : MkCore.{u}) : TopCat.GlueData where
  J := h.J
  U := h.U
  V i := (Opens.toTopCat _).obj (h.V i.1 i.2)
  f i j := (h.V i j).inclusion
  f_id i := by
    -- Porting note: added `dsimp only`
    dsimp only
    exact (h.V_id i).symm ▸ IsIso.of_iso (Opens.inclusionTopIso (h.U i))
  f_open := fun i j : h.J => (h.V i j).openEmbedding
  t := h.t
  t_id i := by ext; rw [h.t_id]; rfl
  t' := h.t'
  t_fac i j k := by
    delta MkCore.t'
    rw [Category.assoc, Category.assoc, pullbackIsoProdSubtype_inv_snd, ← Iso.eq_inv_comp,
      pullbackIsoProdSubtype_inv_fst_assoc]
    ext ⟨⟨⟨x, hx⟩, ⟨x', hx'⟩⟩, rfl : x = x'⟩
    rfl
  cocycle i j k := by
    delta MkCore.t'
    simp_rw [← Category.assoc]
    rw [Iso.comp_inv_eq]
    simp only [Iso.inv_hom_id_assoc, Category.assoc, Category.id_comp]
    

Proof state:
D : GlueData
h : MkCore
i j k : h.J
⊢ ((pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h i j)) (Opens.inclusion (MkCore.V h i k))).hom ≫
      (ContinuousMap.mk fun x =>
          {
            val :=
              ({ val := ↑((MkCore.t h i j) (↑x).1), property := (_ : ↑((MkCore.t h i j) (↑x).1) ∈ MkCore.V h j k) },
                (MkCore.t h i j) (↑x).1),
            property :=
              (_ :
                (Opens.inclusion (MkCore.V h j k))
                    ({ val := ↑((MkCore.t h i j) (↑x).1),
                          property := (_ : ↑((MkCore.t h i j) (↑x).1) ∈ MkCore.V h j k) },
                        (MkCore.t h i j) (↑x).1).1 =
                  (Opens.inclusion (MkCore.V h j k))
                    ({ val := ↑((MkCore.t h i j) (↑x).1),
                          property := (_ : ↑((MkCore.t h i j) (↑x).1) ∈ MkCore.V h j k) },
                        (MkCore.t h i j) (↑x).1).1) }) ≫
        (ContinuousMap.mk fun x =>
            {
              val :=
                ({ val := ↑((MkCore.t h j k) (↑x).1), property := (_ : ↑((MkCore.t h j k) (↑x).1) ∈ MkCore.V h k i) },
                  (MkCore.t h j k) (↑x).1),
              property :=
                (_ :
                  (Opens.inclusion (MkCore.V h k i))
                      ({ val := ↑((MkCore.t h j k) (↑x).1),
                            property := (_ : ↑((MkCore.t h j k) (↑x).1) ∈ MkCore.V h k i) },
                          (MkCore.t h j k) (↑x).1).1 =
                    (Opens.inclusion (MkCore.V h k i))
                      ({ val := ↑((MkCore.t h j k) (↑x).1),
                            property := (_ : ↑((MkCore.t h j k) (↑x).1) ∈ MkCore.V h k i) },
                          (MkCore.t h j k) (↑x).1).1) }) ≫
          ContinuousMap.mk fun x =>
            {
              val :=
                ({ val := ↑((MkCore.t h k i) (↑x).1), property := (_ : ↑((MkCore.t h k i) (↑x).1) ∈ MkCore.V h i j) },
                  (MkCore.t h k i) (↑x).1),
              property :=
                (_ :
                  (Opens.inclusion (MkCore.V h i j))
                      ({ val := ↑((MkCore.t h k i) (↑x).1),
                            property := (_ : ↑((MkCore.t h k i) (↑x).1) ∈ MkCore.V h i j) },
                          (MkCore.t h k i) (↑x).1).1 =
                    (Opens.inclusion (MkCore.V h i j))
                      ({ val := ↑((MkCore.t h k i) (↑x).1),
                            property := (_ : ↑((MkCore.t h k i) (↑x).1) ∈ MkCore.V h i j) },
                          (MkCore.t h k i) (↑x).1).1) }) =
    (pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h i j)) (Opens.inclusion (MkCore.V h i k))).hom

Next tactic:
rw [← Iso.eq_inv_comp, Iso.inv_hom_id]