l3lab/ntp-mathlib huggingface dataset:
theorem opensImagePreimageMap_app' (i j k : D.J) (U : Opens (D.U i).carrier) :
    ∃ eq,
      D.opensImagePreimageMap i j U ≫ (D.f j k).c.app _ =
        ((π₁ j, i, k) ≫ D.t j i ≫ D.f i j).c.app (op U) ≫
          (π₂⁻¹ j, i, k) (unop _) ≫ (D.V (j, k)).presheaf.map (eqToHom eq) := by
  constructor
  delta opensImagePreimageMap
  simp_rw [Category.assoc]
  

Proof state:
case h
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i j k : D.J
U : Opens ↑↑(CategoryTheory.GlueData.U D.toGlueData i)
⊢ (f D.toGlueData i j).c.app (op U) ≫
      (t D.toGlueData j i).c.app ((Opens.map (f D.toGlueData i j).base).op.obj (op U)) ≫
        IsOpenImmersion.invApp (_ : IsOpenImmersion (f D.toGlueData j i))
            { carrier := ⇑(t D.toGlueData j i).base ⁻¹' ↑((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop,
              is_open' :=
                (_ :
                  IsOpen
                    (⇑(t D.toGlueData j i).base ⁻¹' ↑((Opens.map (f D.toGlueData i j).base).op.obj (op U)).unop)) } ≫
          (CategoryTheory.GlueData.U D.toGlueData j).presheaf.map
              (eqToHom
                  (_ :
                    (Opens.map (ι D.toGlueData j).base).obj
                        ((IsOpenMap.functor (_ : IsOpenMap ⇑(ι D.toGlueData i).base)).obj U) =
                      (IsOpenImmersion.openFunctor (_ : IsOpenImmersion (f D.toGlueData j i))).obj
                        ((Opens.map (t D.toGlueData j i).base).obj ((Opens.map (f D.toGlueData i j).base).obj U)))).op ≫
            (f D.toGlueData j k).c.app
              (op
                ((Opens.map (ι D.toGlueData j).base).obj
                  ((IsOpenMap.functor (_ : IsOpenMap ⇑(ι D.toGlueData i).base)).obj U))) =
    (pullback.fst ≫ t D.toGlueData j i ≫ f D.toGlueData i j).c.app (op U) ≫
      IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)
          { carrier := ⇑(pullback.fst ≫ t D.toGlueData j i ≫ f D.toGlueData i j).base ⁻¹' ↑(op U).unop,
            is_open' :=
              (_ : IsOpen (⇑(pullback.fst ≫ t D.toGlueData j i ≫ f D.toGlueData i j).base ⁻¹' ↑(op U).unop)) } ≫
        (V D.toGlueData (j, k)).presheaf.map (eqToHom ?w)
case w
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i j k : D.J
U : Opens ↑↑(CategoryTheory.GlueData.U D.toGlueData i)
⊢ op
      ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj
        {
            unop :=
              { carrier := ⇑(pullback.fst ≫ t D.toGlueData j i ≫ f D.toGlueData i j).base ⁻¹' ↑(op U).unop,
                is_open' :=
                  (_ :
                    IsOpen
                      (⇑(pullback.fst ≫ t D.toGlueData j i ≫ f D.toGlueData i j).base ⁻¹' ↑(op U).unop)) } }.unop) =
    (Opens.map (f D.toGlueData j k).base).op.obj
      (op
        ((Opens.map (ι D.toGlueData j).base).obj ((IsOpenMap.functor (_ : IsOpenMap ⇑(ι D.toGlueData i).base)).obj U)))

Next tactic:
rw [(D.f j k).c.naturality, f_invApp_f_app_assoc]
theorem opensImagePreimageMap_app_assoc (i j k : D.J) (U : Opens (D.U i).carrier) {X' : C}
    (f' : _ ⟶ X') :
    D.opensImagePreimageMap i j U ≫ (D.f j k).c.app _ ≫ f' =
      ((π₁ j, i, k) ≫ D.t j i ≫ D.f i j).c.app (op U) ≫
        (π₂⁻¹ j, i, k) (unop _) ≫
          (D.V (j, k)).presheaf.map
            (eqToHom (opensImagePreimageMap_app' D i j k U).choose) ≫ f' := by
  

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i j k : D.J
U : Opens ↑↑(CategoryTheory.GlueData.U D.toGlueData i)
X' : C
f' :
  ((f D.toGlueData j k).base _* (V D.toGlueData (j, k)).presheaf).obj
      (op
        ((Opens.map (ι D.toGlueData j).base).obj
          ((IsOpenMap.functor (_ : IsOpenMap ⇑(ι D.toGlueData i).base)).obj U))) ⟶
    X'
⊢ opensImagePreimageMap D i j U ≫
      (f D.toGlueData j k).c.app
          (op
            ((Opens.map (ι D.toGlueData j).base).obj
              ((IsOpenMap.functor (_ : IsOpenMap ⇑(ι D.toGlueData i).base)).obj U))) ≫
        f' =
    (pullback.fst ≫ t D.toGlueData j i ≫ f D.toGlueData i j).c.app (op U) ≫
      IsOpenImmersion.invApp (_ : IsOpenImmersion pullback.snd)
          {
              unop :=
                { carrier := ⇑(pullback.fst ≫ t D.toGlueData j i ≫ f D.toGlueData i j).base ⁻¹' ↑(op U).unop,
                  is_open' :=
                    (_ :
                      IsOpen
                        (⇑(pullback.fst ≫ t D.toGlueData j i ≫ f D.toGlueData i j).base ⁻¹' ↑(op U).unop)) } }.unop ≫
        (V D.toGlueData (j, k)).presheaf.map
            (eqToHom
              (_ :
                op
                    ((IsOpenImmersion.openFunctor (_ : IsOpenImmersion pullback.snd)).obj
                      {
                          unop :=
                            {
                              carrier :=
                                ⇑(pullback.fst ≫ t D.toGlueData j i ≫ f D.toGlueData i j).base ⁻¹' ↑(op U).unop,
                              is_open' :=
                                (_ :
                                  IsOpen
                                    (⇑(pullback.fst ≫ t D.toGlueData j i ≫ f D.toGlueData i j).base ⁻¹'
                                      ↑(op U).unop)) } }.unop) =
                  (Opens.map (f D.toGlueData j k).base).op.obj
                    (op
                      ((Opens.map (ι D.toGlueData j).base).obj
                        ((IsOpenMap.functor (_ : IsOpenMap ⇑(ι D.toGlueData i).base)).obj U))))) ≫
          f'

Next tactic:
simpa only [Category.assoc] using congr_arg (· ≫ f') (opensImagePreimageMap_app D i j k U)
/-- (Implementation) We construct the map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_V, U_V)` for each `V` in the gluing
diagram. We will lift these maps into `ιInvApp`. -/
def ιInvAppπApp {i : D.J} (U : Opens (D.U i).carrier) (j) :
    (𝖣.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op j) := by
  

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i : D.J
U : Opens ↑↑(CategoryTheory.GlueData.U D.toGlueData i)
j : WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom
⊢ (CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U) ⟶ (diagramOverOpen D U).obj (op j)

Next tactic:
rcases j with (⟨j, k⟩ | j)
/-- (Implementation) We construct the map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_V, U_V)` for each `V` in the gluing
diagram. We will lift these maps into `ιInvApp`. -/
def ιInvAppπApp {i : D.J} (U : Opens (D.U i).carrier) (j) :
    (𝖣.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op j) := by
  rcases j with (⟨j, k⟩ | j)
  · 

Proof state:
case left.mk
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i : D.J
U : Opens ↑↑(CategoryTheory.GlueData.U D.toGlueData i)
j k : D.J
⊢ (CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U) ⟶
    (diagramOverOpen D U).obj (op (WalkingMultispan.left (j, k)))

Next tactic:
refine'
      D.opensImagePreimageMap i j U ≫ (D.f j k).c.app _ ≫ (D.V (j, k)).presheaf.map (eqToHom _)
/-- (Implementation) We construct the map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_V, U_V)` for each `V` in the gluing
diagram. We will lift these maps into `ιInvApp`. -/
def ιInvAppπApp {i : D.J} (U : Opens (D.U i).carrier) (j) :
    (𝖣.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op j) := by
  rcases j with (⟨j, k⟩ | j)
  · refine'
      D.opensImagePreimageMap i j U ≫ (D.f j k).c.app _ ≫ (D.V (j, k)).presheaf.map (eqToHom _)
    

Proof state:
case left.mk
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i : D.J
U : Opens ↑↑(CategoryTheory.GlueData.U D.toGlueData i)
j k : D.J
⊢ (Opens.map (f D.toGlueData j k).base).op.obj
      (op
        ((Opens.map (ι D.toGlueData j).base).obj
          ((IsOpenMap.functor (_ : IsOpenMap ⇑(ι D.toGlueData i).base)).obj U))) =
    op
      ((Opens.map
            (colimit.ι (MultispanIndex.multispan (diagram D.toGlueData))
                (op (WalkingMultispan.left (j, k))).unop).base).obj
        ((IsOpenMap.functor (_ : IsOpenMap ⇑(ι D.toGlueData i).base)).obj U))

Next tactic:
rw [Functor.op_obj]
/-- (Implementation) We construct the map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_V, U_V)` for each `V` in the gluing
diagram. We will lift these maps into `ιInvApp`. -/
def ιInvAppπApp {i : D.J} (U : Opens (D.U i).carrier) (j) :
    (𝖣.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op j) := by
  rcases j with (⟨j, k⟩ | j)
  · refine'
      D.opensImagePreimageMap i j U ≫ (D.f j k).c.app _ ≫ (D.V (j, k)).presheaf.map (eqToHom _)
    rw [Functor.op_obj]
    

Proof state:
case left.mk
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i : D.J
U : Opens ↑↑(CategoryTheory.GlueData.U D.toGlueData i)
j k : D.J
⊢ op
      ((Opens.map (f D.toGlueData j k).base).obj
        (op
            ((Opens.map (ι D.toGlueData j).base).obj
              ((IsOpenMap.functor (_ : IsOpenMap ⇑(ι D.toGlueData i).base)).obj U))).unop) =
    op
      ((Opens.map
            (colimit.ι (MultispanIndex.multispan (diagram D.toGlueData))
                (op (WalkingMultispan.left (j, k))).unop).base).obj
        ((IsOpenMap.functor (_ : IsOpenMap ⇑(ι D.toGlueData i).base)).obj U))

Next tactic:
congr 1
/-- (Implementation) We construct the map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_V, U_V)` for each `V` in the gluing
diagram. We will lift these maps into `ιInvApp`. -/
def ιInvAppπApp {i : D.J} (U : Opens (D.U i).carrier) (j) :
    (𝖣.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op j) := by
  rcases j with (⟨j, k⟩ | j)
  · refine'
      D.opensImagePreimageMap i j U ≫ (D.f j k).c.app _ ≫ (D.V (j, k)).presheaf.map (eqToHom _)
    rw [Functor.op_obj]
    congr 1; 

Proof state:
case left.mk.e_x
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i : D.J
U : Opens ↑↑(CategoryTheory.GlueData.U D.toGlueData i)
j k : D.J
⊢ (Opens.map (f D.toGlueData j k).base).obj
      (op
          ((Opens.map (ι D.toGlueData j).base).obj
            ((IsOpenMap.functor (_ : IsOpenMap ⇑(ι D.toGlueData i).base)).obj U))).unop =
    (Opens.map
          (colimit.ι (MultispanIndex.multispan (diagram D.toGlueData))
              (op (WalkingMultispan.left (j, k))).unop).base).obj
      ((IsOpenMap.functor (_ : IsOpenMap ⇑(ι D.toGlueData i).base)).obj U)

Next tactic:
ext1
/-- (Implementation) We construct the map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_V, U_V)` for each `V` in the gluing
diagram. We will lift these maps into `ιInvApp`. -/
def ιInvAppπApp {i : D.J} (U : Opens (D.U i).carrier) (j) :
    (𝖣.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op j) := by
  rcases j with (⟨j, k⟩ | j)
  · refine'
      D.opensImagePreimageMap i j U ≫ (D.f j k).c.app _ ≫ (D.V (j, k)).presheaf.map (eqToHom _)
    rw [Functor.op_obj]
    congr 1; ext1
    

Proof state:
case left.mk.e_x.h
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i : D.J
U : Opens ↑↑(CategoryTheory.GlueData.U D.toGlueData i)
j k : D.J
⊢ ↑((Opens.map (f D.toGlueData j k).base).obj
        (op
            ((Opens.map (ι D.toGlueData j).base).obj
              ((IsOpenMap.functor (_ : IsOpenMap ⇑(ι D.toGlueData i).base)).obj U))).unop) =
    ↑((Opens.map
            (colimit.ι (MultispanIndex.multispan (diagram D.toGlueData))
                (op (WalkingMultispan.left (j, k))).unop).base).obj
        ((IsOpenMap.functor (_ : IsOpenMap ⇑(ι D.toGlueData i).base)).obj U))

Next tactic:
dsimp only [Functor.op_obj, Opens.map_coe, unop_op, IsOpenMap.functor_obj_coe]
/-- (Implementation) We construct the map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_V, U_V)` for each `V` in the gluing
diagram. We will lift these maps into `ιInvApp`. -/
def ιInvAppπApp {i : D.J} (U : Opens (D.U i).carrier) (j) :
    (𝖣.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op j) := by
  rcases j with (⟨j, k⟩ | j)
  · refine'
      D.opensImagePreimageMap i j U ≫ (D.f j k).c.app _ ≫ (D.V (j, k)).presheaf.map (eqToHom _)
    rw [Functor.op_obj]
    congr 1; ext1
    dsimp only [Functor.op_obj, Opens.map_coe, unop_op, IsOpenMap.functor_obj_coe]
    

Proof state:
case left.mk.e_x.h
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i : D.J
U : Opens ↑↑(CategoryTheory.GlueData.U D.toGlueData i)
j k : D.J
⊢ ⇑(f D.toGlueData j k).base ⁻¹' (⇑(ι D.toGlueData j).base ⁻¹' (⇑(ι D.toGlueData i).base '' ↑U)) =
    ⇑(colimit.ι (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.left (j, k))).base ⁻¹'
      (⇑(ι D.toGlueData i).base '' ↑U)

Next tactic:
rw [Set.preimage_preimage]
/-- (Implementation) We construct the map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_V, U_V)` for each `V` in the gluing
diagram. We will lift these maps into `ιInvApp`. -/
def ιInvAppπApp {i : D.J} (U : Opens (D.U i).carrier) (j) :
    (𝖣.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op j) := by
  rcases j with (⟨j, k⟩ | j)
  · refine'
      D.opensImagePreimageMap i j U ≫ (D.f j k).c.app _ ≫ (D.V (j, k)).presheaf.map (eqToHom _)
    rw [Functor.op_obj]
    congr 1; ext1
    dsimp only [Functor.op_obj, Opens.map_coe, unop_op, IsOpenMap.functor_obj_coe]
    rw [Set.preimage_preimage]
    

Proof state:
case left.mk.e_x.h
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i : D.J
U : Opens ↑↑(CategoryTheory.GlueData.U D.toGlueData i)
j k : D.J
⊢ (fun x => (ι D.toGlueData j).base ((f D.toGlueData j k).base x)) ⁻¹' (⇑(ι D.toGlueData i).base '' ↑U) =
    ⇑(colimit.ι (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.left (j, k))).base ⁻¹'
      (⇑(ι D.toGlueData i).base '' ↑U)

Next tactic:
change (D.f j k ≫ 𝖣.ι j).base ⁻¹' _ = _
/-- (Implementation) We construct the map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_V, U_V)` for each `V` in the gluing
diagram. We will lift these maps into `ιInvApp`. -/
def ιInvAppπApp {i : D.J} (U : Opens (D.U i).carrier) (j) :
    (𝖣.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op j) := by
  rcases j with (⟨j, k⟩ | j)
  · refine'
      D.opensImagePreimageMap i j U ≫ (D.f j k).c.app _ ≫ (D.V (j, k)).presheaf.map (eqToHom _)
    rw [Functor.op_obj]
    congr 1; ext1
    dsimp only [Functor.op_obj, Opens.map_coe, unop_op, IsOpenMap.functor_obj_coe]
    rw [Set.preimage_preimage]
    change (D.f j k ≫ 𝖣.ι j).base ⁻¹' _ = _
    -- Porting note : used to be `congr 3`
    

Proof state:
case left.mk.e_x.h
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i : D.J
U : Opens ↑↑(CategoryTheory.GlueData.U D.toGlueData i)
j k : D.J
⊢ ⇑(f D.toGlueData j k ≫ ι D.toGlueData j).base ⁻¹' (⇑(ι D.toGlueData i).base '' ↑U) =
    ⇑(colimit.ι (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.left (j, k))).base ⁻¹'
      (⇑(ι D.toGlueData i).base '' ↑U)

Next tactic:
refine congr_arg (· ⁻¹' _) ?_
/-- (Implementation) We construct the map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_V, U_V)` for each `V` in the gluing
diagram. We will lift these maps into `ιInvApp`. -/
def ιInvAppπApp {i : D.J} (U : Opens (D.U i).carrier) (j) :
    (𝖣.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op j) := by
  rcases j with (⟨j, k⟩ | j)
  · refine'
      D.opensImagePreimageMap i j U ≫ (D.f j k).c.app _ ≫ (D.V (j, k)).presheaf.map (eqToHom _)
    rw [Functor.op_obj]
    congr 1; ext1
    dsimp only [Functor.op_obj, Opens.map_coe, unop_op, IsOpenMap.functor_obj_coe]
    rw [Set.preimage_preimage]
    change (D.f j k ≫ 𝖣.ι j).base ⁻¹' _ = _
    -- Porting note : used to be `congr 3`
    refine congr_arg (· ⁻¹' _) ?_
    

Proof state:
case left.mk.e_x.h
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i : D.J
U : Opens ↑↑(CategoryTheory.GlueData.U D.toGlueData i)
j k : D.J
⊢ ⇑(f D.toGlueData j k ≫ ι D.toGlueData j).base =
    ⇑(colimit.ι (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.left (j, k))).base

Next tactic:
convert congr_arg (ContinuousMap.toFun (α := D.V ⟨j, k⟩) (β := D.glued) ·) ?_
/-- (Implementation) We construct the map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_V, U_V)` for each `V` in the gluing
diagram. We will lift these maps into `ιInvApp`. -/
def ιInvAppπApp {i : D.J} (U : Opens (D.U i).carrier) (j) :
    (𝖣.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op j) := by
  rcases j with (⟨j, k⟩ | j)
  · refine'
      D.opensImagePreimageMap i j U ≫ (D.f j k).c.app _ ≫ (D.V (j, k)).presheaf.map (eqToHom _)
    rw [Functor.op_obj]
    congr 1; ext1
    dsimp only [Functor.op_obj, Opens.map_coe, unop_op, IsOpenMap.functor_obj_coe]
    rw [Set.preimage_preimage]
    change (D.f j k ≫ 𝖣.ι j).base ⁻¹' _ = _
    -- Porting note : used to be `congr 3`
    refine congr_arg (· ⁻¹' _) ?_
    convert congr_arg (ContinuousMap.toFun (α := D.V ⟨j, k⟩) (β := D.glued) ·) ?_
    

Proof state:
case left.mk.e_x.h.convert_3
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i : D.J
U : Opens ↑↑(CategoryTheory.GlueData.U D.toGlueData i)
j k : D.J
⊢ (f D.toGlueData j k ≫ ι D.toGlueData j).base =
    (colimit.ι (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.left (j, k))).base

Next tactic:
refine congr_arg (PresheafedSpace.Hom.base (C := C) ·) ?_
/-- (Implementation) We construct the map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_V, U_V)` for each `V` in the gluing
diagram. We will lift these maps into `ιInvApp`. -/
def ιInvAppπApp {i : D.J} (U : Opens (D.U i).carrier) (j) :
    (𝖣.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op j) := by
  rcases j with (⟨j, k⟩ | j)
  · refine'
      D.opensImagePreimageMap i j U ≫ (D.f j k).c.app _ ≫ (D.V (j, k)).presheaf.map (eqToHom _)
    rw [Functor.op_obj]
    congr 1; ext1
    dsimp only [Functor.op_obj, Opens.map_coe, unop_op, IsOpenMap.functor_obj_coe]
    rw [Set.preimage_preimage]
    change (D.f j k ≫ 𝖣.ι j).base ⁻¹' _ = _
    -- Porting note : used to be `congr 3`
    refine congr_arg (· ⁻¹' _) ?_
    convert congr_arg (ContinuousMap.toFun (α := D.V ⟨j, k⟩) (β := D.glued) ·) ?_
    refine congr_arg (PresheafedSpace.Hom.base (C := C) ·) ?_
    

Proof state:
case left.mk.e_x.h.convert_3
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i : D.J
U : Opens ↑↑(CategoryTheory.GlueData.U D.toGlueData i)
j k : D.J
⊢ f D.toGlueData j k ≫ ι D.toGlueData j =
    colimit.ι (MultispanIndex.multispan (diagram D.toGlueData)) (WalkingMultispan.left (j, k))

Next tactic:
exact colimit.w 𝖣.diagram.multispan (WalkingMultispan.Hom.fst (j, k))
/-- (Implementation) We construct the map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_V, U_V)` for each `V` in the gluing
diagram. We will lift these maps into `ιInvApp`. -/
def ιInvAppπApp {i : D.J} (U : Opens (D.U i).carrier) (j) :
    (𝖣.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op j) := by
  rcases j with (⟨j, k⟩ | j)
  · refine'
      D.opensImagePreimageMap i j U ≫ (D.f j k).c.app _ ≫ (D.V (j, k)).presheaf.map (eqToHom _)
    rw [Functor.op_obj]
    congr 1; ext1
    dsimp only [Functor.op_obj, Opens.map_coe, unop_op, IsOpenMap.functor_obj_coe]
    rw [Set.preimage_preimage]
    change (D.f j k ≫ 𝖣.ι j).base ⁻¹' _ = _
    -- Porting note : used to be `congr 3`
    refine congr_arg (· ⁻¹' _) ?_
    convert congr_arg (ContinuousMap.toFun (α := D.V ⟨j, k⟩) (β := D.glued) ·) ?_
    refine congr_arg (PresheafedSpace.Hom.base (C := C) ·) ?_
    exact colimit.w 𝖣.diagram.multispan (WalkingMultispan.Hom.fst (j, k))
  · 

Proof state:
case right
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i : D.J
U : Opens ↑↑(CategoryTheory.GlueData.U D.toGlueData i)
j : (diagram D.toGlueData).R
⊢ (CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U) ⟶
    (diagramOverOpen D U).obj (op (WalkingMultispan.right j))

Next tactic:
exact D.opensImagePreimageMap i j U
set_option maxHeartbeats 600000 in
/-- (Implementation) The natural map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_X, 𝖣.ι i '' U)`.
This forms the inverse of `(𝖣.ι i).c.app (op U)`. -/
def ιInvApp {i : D.J} (U : Opens (D.U i).carrier) :
    (D.U i).presheaf.obj (op U) ⟶ limit (D.diagramOverOpen U) :=
  limit.lift (D.diagramOverOpen U)
    { pt := (D.U i).presheaf.obj (op U)
      π :=
        { app := fun j => D.ιInvAppπApp U (unop j)
          naturality := fun {X Y} f' => by
            

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i : D.J
U : Opens ↑↑(CategoryTheory.GlueData.U D.toGlueData i)
X Y : (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)ᵒᵖ
f' : X ⟶ Y
⊢ ((Functor.const (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)ᵒᵖ).obj
            ((CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U))).map
        f' ≫
      (fun j => ιInvAppπApp D U j.unop) Y =
    (fun j => ιInvAppπApp D U j.unop) X ≫ (diagramOverOpen D U).map f'

Next tactic:
induction X using Opposite.rec' with | h X => ?_
set_option maxHeartbeats 600000 in
/-- (Implementation) The natural map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_X, 𝖣.ι i '' U)`.
This forms the inverse of `(𝖣.ι i).c.app (op U)`. -/
def ιInvApp {i : D.J} (U : Opens (D.U i).carrier) :
    (D.U i).presheaf.obj (op U) ⟶ limit (D.diagramOverOpen U) :=
  limit.lift (D.diagramOverOpen U)
    { pt := (D.U i).presheaf.obj (op U)
      π :=
        { app := fun j => D.ιInvAppπApp U (unop j)
          naturality := fun {X Y} f' => by
            induction X using Opposite.rec' with | h X => ?_
            

Proof state:
case h
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i : D.J
U : Opens ↑↑(CategoryTheory.GlueData.U D.toGlueData i)
Y : (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)ᵒᵖ
X : WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom
f' : op X ⟶ Y
⊢ ((Functor.const (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)ᵒᵖ).obj
            ((CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U))).map
        f' ≫
      (fun j => ιInvAppπApp D U j.unop) Y =
    (fun j => ιInvAppπApp D U j.unop) (op X) ≫ (diagramOverOpen D U).map f'

Next tactic:
induction Y using Opposite.rec' with | h Y => ?_
set_option maxHeartbeats 600000 in
/-- (Implementation) The natural map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_X, 𝖣.ι i '' U)`.
This forms the inverse of `(𝖣.ι i).c.app (op U)`. -/
def ιInvApp {i : D.J} (U : Opens (D.U i).carrier) :
    (D.U i).presheaf.obj (op U) ⟶ limit (D.diagramOverOpen U) :=
  limit.lift (D.diagramOverOpen U)
    { pt := (D.U i).presheaf.obj (op U)
      π :=
        { app := fun j => D.ιInvAppπApp U (unop j)
          naturality := fun {X Y} f' => by
            induction X using Opposite.rec' with | h X => ?_
            induction Y using Opposite.rec' with | h Y => ?_
            

Proof state:
case h.h
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i : D.J
U : Opens ↑↑(CategoryTheory.GlueData.U D.toGlueData i)
X Y : WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom
f' : op X ⟶ op Y
⊢ ((Functor.const (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)ᵒᵖ).obj
            ((CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U))).map
        f' ≫
      (fun j => ιInvAppπApp D U j.unop) (op Y) =
    (fun j => ιInvAppπApp D U j.unop) (op X) ≫ (diagramOverOpen D U).map f'

Next tactic:
let f : Y ⟶ X := f'.unop
set_option maxHeartbeats 600000 in
/-- (Implementation) The natural map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_X, 𝖣.ι i '' U)`.
This forms the inverse of `(𝖣.ι i).c.app (op U)`. -/
def ιInvApp {i : D.J} (U : Opens (D.U i).carrier) :
    (D.U i).presheaf.obj (op U) ⟶ limit (D.diagramOverOpen U) :=
  limit.lift (D.diagramOverOpen U)
    { pt := (D.U i).presheaf.obj (op U)
      π :=
        { app := fun j => D.ιInvAppπApp U (unop j)
          naturality := fun {X Y} f' => by
            induction X using Opposite.rec' with | h X => ?_
            induction Y using Opposite.rec' with | h Y => ?_
            let f : Y ⟶ X := f'.unop; 

Proof state:
case h.h
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i : D.J
U : Opens ↑↑(CategoryTheory.GlueData.U D.toGlueData i)
X Y : WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom
f' : op X ⟶ op Y
f : Y ⟶ X := f'.unop
⊢ ((Functor.const (WalkingMultispan (diagram D.toGlueData).fstFrom (diagram D.toGlueData).sndFrom)ᵒᵖ).obj
            ((CategoryTheory.GlueData.U D.toGlueData i).presheaf.obj (op U))).map
        f' ≫
      (fun j => ιInvAppπApp D U j.unop) (op Y) =
    (fun j => ιInvAppπApp D U j.unop) (op X) ≫ (diagramOverOpen D U).map f'

Next tactic:
have : f' = f.op := rfl