l3lab/ntp-mathlib huggingface dataset:
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
⊢ F.val.map f₁.op (x g₁ hg₁) = F.val.map f₂.op (x g₂ hg₂)

Next tactic:
let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
⊢ F.val.map f₁.op (x g₁ hg₁) = F.val.map f₂.op (x g₂ hg₂)

Next tactic:
let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
⊢ F.val.map f₁.op (x g₁ hg₁) = F.val.map f₂.op (x g₂ hg₂)

Next tactic:
let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by simpa using (Over.forget _).congr_map h.symm =≫ Z.hom)
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by 

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
⊢ f₂.left ≫ Y₂.hom = W'.hom

Next tactic:
simpa using (Over.forget _).congr_map h.symm =≫ Z.hom
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by simpa using (Over.forget _).congr_map h.symm =≫ Z.hom)
    

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
⊢ F.val.map f₁.op (x g₁ hg₁) = F.val.map f₂.op (x g₂ hg₂)

Next tactic:
let e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl _)
      (by simpa using (Over.w f₁).symm)
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by simpa using (Over.forget _).congr_map h.symm =≫ Z.hom)
    let e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl _)
      (by 

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
⊢ (Iso.refl ((Over.map f).obj W').left).hom ≫ W.hom = ((Over.map f).obj W').hom

Next tactic:
simpa using (Over.w f₁).symm
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by simpa using (Over.forget _).congr_map h.symm =≫ Z.hom)
    let e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl _)
      (by simpa using (Over.w f₁).symm)
    

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left)
⊢ F.val.map f₁.op (x g₁ hg₁) = F.val.map f₂.op (x g₂ hg₂)

Next tactic:
convert congr_arg (F.val.map e.inv.op)
      (hx g₁' g₂' hg₁ hg₂ (by ext; exact (Over.forget _).congr_map h)) using 1
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by simpa using (Over.forget _).congr_map h.symm =≫ Z.hom)
    let e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl _)
      (by simpa using (Over.w f₁).symm)
    convert congr_arg (F.val.map e.inv.op)
      (hx g₁' g₂' hg₁ hg₂ (by 

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left)
⊢ g₁' ≫ g₁ = g₂' ≫ g₂

Next tactic:
ext
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by simpa using (Over.forget _).congr_map h.symm =≫ Z.hom)
    let e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl _)
      (by simpa using (Over.w f₁).symm)
    convert congr_arg (F.val.map e.inv.op)
      (hx g₁' g₂' hg₁ hg₂ (by ext; 

Proof state:
case h
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left)
⊢ (g₁' ≫ g₁).left = (g₂' ≫ g₂).left

Next tactic:
exact (Over.forget _).congr_map h
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by simpa using (Over.forget _).congr_map h.symm =≫ Z.hom)
    let e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl _)
      (by simpa using (Over.w f₁).symm)
    convert congr_arg (F.val.map e.inv.op)
      (hx g₁' g₂' hg₁ hg₂ (by ext; exact (Over.forget _).congr_map h)) using 1
    

Proof state:
case h.e'_2
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left)
⊢ F.val.map f₁.op (x g₁ hg₁) = F.val.map e.inv.op (((Over.map f).op ⋙ F.val).map g₁'.op (x g₁ hg₁))
case h.e'_3
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left)
⊢ F.val.map f₂.op (x g₂ hg₂) = F.val.map e.inv.op (((Over.map f).op ⋙ F.val).map g₂'.op (x g₂ hg₂))

Next tactic:
all_goals
      dsimp
      rw [← FunctorToTypes.map_comp_apply]
      apply congr_fun
      congr 1
      rw [← op_comp]
      congr 1
      ext
      simp
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by simpa using (Over.forget _).congr_map h.symm =≫ Z.hom)
    let e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl _)
      (by simpa using (Over.w f₁).symm)
    convert congr_arg (F.val.map e.inv.op)
      (hx g₁' g₂' hg₁ hg₂ (by ext; exact (Over.forget _).congr_map h)) using 1
    all_goals
      

Proof state:
case h.e'_2
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left)
⊢ F.val.map f₁.op (x g₁ hg₁) = F.val.map e.inv.op (((Over.map f).op ⋙ F.val).map g₁'.op (x g₁ hg₁))

Next tactic:
dsimp
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by simpa using (Over.forget _).congr_map h.symm =≫ Z.hom)
    let e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl _)
      (by simpa using (Over.w f₁).symm)
    convert congr_arg (F.val.map e.inv.op)
      (hx g₁' g₂' hg₁ hg₂ (by ext; exact (Over.forget _).congr_map h)) using 1
    all_goals
      dsimp
      

Proof state:
case h.e'_2
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left)
⊢ F.val.map f₁.op (x g₁ hg₁) =
    F.val.map (Over.isoMk (Iso.refl W.left)).inv.op (F.val.map ((Over.map f).map (Over.homMk f₁.left)).op (x g₁ hg₁))

Next tactic:
rw [← FunctorToTypes.map_comp_apply]
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by simpa using (Over.forget _).congr_map h.symm =≫ Z.hom)
    let e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl _)
      (by simpa using (Over.w f₁).symm)
    convert congr_arg (F.val.map e.inv.op)
      (hx g₁' g₂' hg₁ hg₂ (by ext; exact (Over.forget _).congr_map h)) using 1
    all_goals
      dsimp
      rw [← FunctorToTypes.map_comp_apply]
      

Proof state:
case h.e'_2
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left)
⊢ F.val.map f₁.op (x g₁ hg₁) =
    F.val.map (((Over.map f).map (Over.homMk f₁.left)).op ≫ (Over.isoMk (Iso.refl W.left)).inv.op) (x g₁ hg₁)

Next tactic:
apply congr_fun
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by simpa using (Over.forget _).congr_map h.symm =≫ Z.hom)
    let e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl _)
      (by simpa using (Over.w f₁).symm)
    convert congr_arg (F.val.map e.inv.op)
      (hx g₁' g₂' hg₁ hg₂ (by ext; exact (Over.forget _).congr_map h)) using 1
    all_goals
      dsimp
      rw [← FunctorToTypes.map_comp_apply]
      apply congr_fun
      

Proof state:
case h.e'_2.h
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left)
⊢ F.val.map f₁.op = F.val.map (((Over.map f).map (Over.homMk f₁.left)).op ≫ (Over.isoMk (Iso.refl W.left)).inv.op)

Next tactic:
congr 1
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by simpa using (Over.forget _).congr_map h.symm =≫ Z.hom)
    let e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl _)
      (by simpa using (Over.w f₁).symm)
    convert congr_arg (F.val.map e.inv.op)
      (hx g₁' g₂' hg₁ hg₂ (by ext; exact (Over.forget _).congr_map h)) using 1
    all_goals
      dsimp
      rw [← FunctorToTypes.map_comp_apply]
      apply congr_fun
      congr 1
      

Proof state:
case h.e'_2.h.e_a
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left)
⊢ f₁.op = ((Over.map f).map (Over.homMk f₁.left)).op ≫ (Over.isoMk (Iso.refl W.left)).inv.op

Next tactic:
rw [← op_comp]
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by simpa using (Over.forget _).congr_map h.symm =≫ Z.hom)
    let e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl _)
      (by simpa using (Over.w f₁).symm)
    convert congr_arg (F.val.map e.inv.op)
      (hx g₁' g₂' hg₁ hg₂ (by ext; exact (Over.forget _).congr_map h)) using 1
    all_goals
      dsimp
      rw [← FunctorToTypes.map_comp_apply]
      apply congr_fun
      congr 1
      rw [← op_comp]
      

Proof state:
case h.e'_2.h.e_a
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left)
⊢ f₁.op = ((Over.isoMk (Iso.refl W.left)).inv ≫ (Over.map f).map (Over.homMk f₁.left)).op

Next tactic:
congr 1
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by simpa using (Over.forget _).congr_map h.symm =≫ Z.hom)
    let e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl _)
      (by simpa using (Over.w f₁).symm)
    convert congr_arg (F.val.map e.inv.op)
      (hx g₁' g₂' hg₁ hg₂ (by ext; exact (Over.forget _).congr_map h)) using 1
    all_goals
      dsimp
      rw [← FunctorToTypes.map_comp_apply]
      apply congr_fun
      congr 1
      rw [← op_comp]
      congr 1
      

Proof state:
case h.e'_2.h.e_a.e_f
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left)
⊢ f₁ = (Over.isoMk (Iso.refl W.left)).inv ≫ (Over.map f).map (Over.homMk f₁.left)

Next tactic:
ext
lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
    CompatiblePreserving (J.over Y) (Over.map f) where
  compatible {F Z T x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} := by
    let W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
    let g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
    let g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
      (by simpa using (Over.forget _).congr_map h.symm =≫ Z.hom)
    let e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl _)
      (by simpa using (Over.w f₁).symm)
    convert congr_arg (F.val.map e.inv.op)
      (hx g₁' g₂' hg₁ hg₂ (by ext; exact (Over.forget _).congr_map h)) using 1
    all_goals
      dsimp
      rw [← FunctorToTypes.map_comp_apply]
      apply congr_fun
      congr 1
      rw [← op_comp]
      congr 1
      ext
      

Proof state:
case h.e'_2.h.e_a.e_f.h
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (over J Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : Presieve.FamilyOfElements.Compatible x
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left)
⊢ f₁.left = ((Over.isoMk (Iso.refl W.left)).inv ≫ (Over.map f).map (Over.homMk f₁.left)).left

Next tactic:
simp