-- Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit
Goal:
I : Type u
inst : Category.{u, u} I
S X : Scheme
D : I ⥤ Scheme
t : D ⟶ Functor.obj (Functor.const I) S
f : X ⟶ S
c : Cone D
hc : IsLimit c
inst_1 : IsCofiltered I
inst_2 : LocallyOfFinitePresentation f
inst_3 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst_4 : ∀ (i : I), CompactSpace.{u} ↥(D.obj i)
inst_5 : ∀ (i : I), QuasiSeparatedSpace.{u} ↥(D.obj i)
a : c.pt ⟶ X
ha : c.π ≫ t = CategoryTheory.Functor.map (Functor.const I) (a ≫ f)
𝒰 :
  IsOpenCover (X := ↥c.pt)
    fun
      (V :
        { x :
          TopologicalSpace.Opens ↥c.pt ×
            { x : TopologicalSpace.Opens ↥X × (↑S.affineOpens : Type u) //
              x.1 ∈ X.affineOpens ∧
                x.1 ≤
                  (Opens.comap (TopCat.Hom.hom f.base) : (a : TopologicalSpace.Opens ↥S) → TopologicalSpace.Opens ↥X)
                    (↑x.2 : TopologicalSpace.Opens ↥S) } //
          Membership.mem (γ := Set (TopologicalSpace.Opens ↥c.pt)) c.pt.affineOpens x.1 ∧
            x.1 ≤
              (Opens.comap (TopCat.Hom.hom a.base) : (a : TopologicalSpace.Opens ↥X) → TopologicalSpace.Opens ↥c.pt)
                (↑x.2 : TopologicalSpace.Opens ↥X × (↑S.affineOpens : Type u)).1 }) ↦
    (↑V :
        TopologicalSpace.Opens ↥c.pt ×
          { x : TopologicalSpace.Opens ↥X × (↑S.affineOpens : Type u) //
            x.1 ∈ X.affineOpens ∧
              x.1 ≤
                (Opens.comap (TopCat.Hom.hom f.base) : (a : TopologicalSpace.Opens ↥S) → TopologicalSpace.Opens ↥X)
                  (↑x.2 : TopologicalSpace.Opens ↥S) }).1
i : I
s :
  Finset
    { x :
      TopologicalSpace.Opens ↥c.pt ×
        { x : TopologicalSpace.Opens ↥X × (↑S.affineOpens : Type u) //
          x.1 ∈ X.affineOpens ∧
            x.1 ≤
              (Opens.comap (TopCat.Hom.hom f.base) : (a : TopologicalSpace.Opens ↥S) → TopologicalSpace.Opens ↥X)
                (↑x.2 : TopologicalSpace.Opens ↥S) } //
      Membership.mem (γ := Set (TopologicalSpace.Opens ↥c.pt)) c.pt.affineOpens x.1 ∧
        x.1 ≤
          (Opens.comap (TopCat.Hom.hom a.base) : (a : TopologicalSpace.Opens ↥X) → TopologicalSpace.Opens ↥c.pt)
            (↑x.2 : TopologicalSpace.Opens ↥X × (↑S.affineOpens : Type u)).1 }
𝒱 : ↥s → (D.obj i).Opens
h𝒱 : IsOpenCover 𝒱
h𝒱𝒰 :
  ∀ (j : ↥s),
    IsAffineOpen (𝒱 j) ∧
      (↑(↑j :
                { x :
                  TopologicalSpace.Opens ↥c.pt ×
                    { x : TopologicalSpace.Opens ↥X × (↑S.affineOpens : Type u) //
                      x.1 ∈ X.affineOpens ∧
                        x.1 ≤
                          (Opens.comap (TopCat.Hom.hom f.base) :
                              (a : TopologicalSpace.Opens ↥S) → TopologicalSpace.Opens ↥X)
                            (↑x.2 : TopologicalSpace.Opens ↥S) } //
                  x.1 ∈ c.pt.affineOpens ∧
                    x.1 ≤
                      (Opens.comap (TopCat.Hom.hom a.base) :
                          (a : TopologicalSpace.Opens ↥X) → TopologicalSpace.Opens ↥c.pt)
                        (↑x.2 : TopologicalSpace.Opens ↥X × (↑S.affineOpens : Type u)).1 }) :
            TopologicalSpace.Opens ↥c.pt ×
              { x : TopologicalSpace.Opens ↥X × (↑S.affineOpens : Type u) //
                x.1 ∈ X.affineOpens ∧
                  x.1 ≤
                    (Opens.comap (TopCat.Hom.hom f.base) : (a : TopologicalSpace.Opens ↥S) → TopologicalSpace.Opens ↥X)
                      (↑x.2 : TopologicalSpace.Opens ↥S) }).1 =
        c.π.app i ⁻¹ᵁ 𝒱 j
i' : I
fi'i : i' ⟶ i
hi' :
  ∀ (j : ↥s),
    D.map fi'i ⁻¹ᵁ 𝒱 j ≤
      t.app i' ⁻¹ᵁ
        (↑(↑(↑(↑j :
                          { x :
                            TopologicalSpace.Opens ↥c.pt ×
                              { x : TopologicalSpace.Opens ↥X × (↑S.affineOpens : Type u) //
                                x.1 ∈ X.affineOpens ∧
                                  x.1 ≤
                                    (Opens.comap (TopCat.Hom.hom f.base) :
                                        (a : TopologicalSpace.Opens ↥S) → TopologicalSpace.Opens ↥X)
                                      (↑x.2 : TopologicalSpace.Opens ↥S) } //
                            x.1 ∈ c.pt.affineOpens ∧
                              x.1 ≤
                                (Opens.comap (TopCat.Hom.hom a.base) :
                                    (a : TopologicalSpace.Opens ↥X) → TopologicalSpace.Opens ↥c.pt)
                                  (↑x.2 : TopologicalSpace.Opens ↥X × (↑S.affineOpens : Type u)).1 }) :
                      TopologicalSpace.Opens ↥c.pt ×
                        { x : TopologicalSpace.Opens ↥X × (↑S.affineOpens : Type u) //
                          x.1 ∈ X.affineOpens ∧
                            x.1 ≤
                              (Opens.comap (TopCat.Hom.hom f.base) :
                                  (a : TopologicalSpace.Opens ↥S) → TopologicalSpace.Opens ↥X)
                                (↑x.2 : TopologicalSpace.Opens ↥S) }).2 :
                TopologicalSpace.Opens ↥X × (↑S.affineOpens : Type u)).2 :
          TopologicalSpace.Opens ↥S)
⊢ ∃ (k : I) (fk : k ⟶ i),
    ∀ (j : ↥s),
      ∃ (ak : (↑(D.map fk ⁻¹ᵁ 𝒱 j) : Scheme) ⟶ X),
        ak ≫ f = (D.map fk ⁻¹ᵁ 𝒱 j).ι ≫ t.app k ∧
          c.π.app k ∣_ D.map fk ⁻¹ᵁ 𝒱 j ≫ ak = (c.π.app k ⁻¹ᵁ D.map fk ⁻¹ᵁ 𝒱 j).ι ≫ a

Tactic:
have : ∃ k, ∃ (fk : k ⟶ i), ∀ j, ∃ (ak : ↑(D.map fk ⁻¹ᵁ 𝒱 j) ⟶ X),
      ak ≫ f = Opens.ι _ ≫ t.app k ∧ c.π.app _ ∣_ _ ≫ ak = Opens.ι _ ≫ a := by
    let 𝒱' := (D.map fi'i ⁻¹ᵁ 𝒱 ·)
    have h𝒱'𝒰 (j : s) : c.π.app i' ⁻¹ᵁ 𝒱' j = j.1.1.1 := by
      rw [← Hom.comp_preimage, c.w fi'i]; exact (h𝒱𝒰 j).2.symm
    have _ (j k) : IsAffine ((opensDiagram D i' (𝒱' j)).obj k) := ((h𝒱𝒰 _).1.preimage _).preimage _
    let t𝒱 (j : _) : opensDiagram D i' (𝒱' j) ⟶ (Functor.const (Over i')).obj j.1.1.2.1.2 :=
    { app k := (t.app k.left).resLE _ _ <| by
        refine (Hom.preimage_mono _ (hi' _)).trans ?_
        simp only [Functor.const_obj_obj, ← Hom.comp_preimage, t.naturality,
          Functor.const_obj_map, Category.comp_id, le_refl]
      naturality {k₁ k₂} f₁₂ := by simp [Hom.resLE_comp_resLE] }
    have (j : s) : IsAffine j.1.1.2.1.1 := j.1.1.2.prop.1
    choose k ak hk hk' using fun j ↦ exists_π_app_comp_eq_of_locallyOfFinitePresentation_of_isAffine
      _ (t𝒱 j) (f.resLE _ _ j.1.1.2.prop.2) _ (isLimitOpensCone D c hc i' (𝒱' j))
      (a.resLE _ _ ((h𝒱'𝒰 _).trans_le j.1.prop.2)) (by
      ext k
      simp [t𝒱, Hom.resLE_comp_resLE, show c.π.app k.left ≫ t.app k.left = a ≫ f from
        congr(($ha).app k.left)])
    obtain ⟨i'', fi''i', fi'', hi''⟩ := IsCofiltered.wideCospan fun j ↦ (k j).hom
    refine ⟨i'', fi''i' ≫ fi'i, fun j ↦
      ⟨Scheme.homOfLE _ ?_ ≫ D.map (fi'' _) ∣_ _ ≫ ak j ≫ Opens.ι _, ?_, ?_⟩⟩
    · simp only [← Hom.comp_preimage, ← Functor.map_comp, 𝒱', reassoc_of% hi'']; rfl
    · have : ak j ≫ Opens.ι _ ≫ f = Opens.ι _ ≫ t.app (k j).left := by
        simpa [t𝒱] using congr($(hk' j) ≫ Opens.ι _)
      simp [this]
    · have e : c.π.app i'' ⁻¹ᵁ D.map (fi''i' ≫ fi'i) ⁻¹ᵁ 𝒱 j ≤ c.π.app i' ⁻¹ᵁ 𝒱' j := by
        simp only [← Hom.comp_preimage, Cone.w, 𝒱']; rfl
      simpa [← AlgebraicGeometry.Scheme.Hom.resLE_eq_morphismRestrict,
        Scheme.Hom.resLE_comp_resLE_assoc] using congr(Scheme.homOfLE _ e ≫ $(hk j) ≫ Opens.ι _)