l3lab/ntp-mathlib huggingface dataset:
@[simp]
theorem getSection_commute {Y Z : StructuredArrow (op U) G.op} (f : Y ⟶ Z) :
    getSection ℱ hS hx Y ≫ ℱ.val.map f.right = getSection ℱ hS hx Z := by
  apply getSection_is_unique
  intro V' fV' hV'
  

Proof state:
case H
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
Y Z : StructuredArrow (op U) G.op
f : Y ⟶ Z
V' : C
fV' : V' ⟶ Z.right.unop
hV' : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'
⊢ ((𝟭 (Cᵒᵖ ⥤ A)).obj ℱ.val ⋙ coyoneda.obj (op X)).map fV'.op (getSection ℱ hS hx Y ≫ ℱ.val.map f.right) =
    pulledbackFamily ℱ S x Z fV' hV'

Next tactic:
have eq : Z.hom = Y.hom ≫ (G.map f.right.unop).op := by
    convert f.w
    erw [Category.id_comp]
@[simp]
theorem getSection_commute {Y Z : StructuredArrow (op U) G.op} (f : Y ⟶ Z) :
    getSection ℱ hS hx Y ≫ ℱ.val.map f.right = getSection ℱ hS hx Z := by
  apply getSection_is_unique
  intro V' fV' hV'
  have eq : Z.hom = Y.hom ≫ (G.map f.right.unop).op := by
    

Proof state:
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
Y Z : StructuredArrow (op U) G.op
f : Y ⟶ Z
V' : C
fV' : V' ⟶ Z.right.unop
hV' : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'
⊢ Z.hom = Y.hom ≫ (G.map f.right.unop).op

Next tactic:
convert f.w
@[simp]
theorem getSection_commute {Y Z : StructuredArrow (op U) G.op} (f : Y ⟶ Z) :
    getSection ℱ hS hx Y ≫ ℱ.val.map f.right = getSection ℱ hS hx Z := by
  apply getSection_is_unique
  intro V' fV' hV'
  have eq : Z.hom = Y.hom ≫ (G.map f.right.unop).op := by
    convert f.w
    

Proof state:
case h.e'_2.h
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
Y Z : StructuredArrow (op U) G.op
f : Y ⟶ Z
V' : C
fV' : V' ⟶ Z.right.unop
hV' : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'
e_1✝ :
  ((Functor.fromPUnit (op U)).obj Z.left ⟶ G.op.obj Z.right) =
    ((Functor.fromPUnit (op U)).obj Y.left ⟶ G.op.obj Z.right)
⊢ Z.hom = (Functor.fromPUnit (op U)).map f.left ≫ Z.hom

Next tactic:
erw [Category.id_comp]
@[simp]
theorem getSection_commute {Y Z : StructuredArrow (op U) G.op} (f : Y ⟶ Z) :
    getSection ℱ hS hx Y ≫ ℱ.val.map f.right = getSection ℱ hS hx Z := by
  apply getSection_is_unique
  intro V' fV' hV'
  have eq : Z.hom = Y.hom ≫ (G.map f.right.unop).op := by
    convert f.w
    erw [Category.id_comp]
  

Proof state:
case H
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
Y Z : StructuredArrow (op U) G.op
f : Y ⟶ Z
V' : C
fV' : V' ⟶ Z.right.unop
hV' : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'
eq : Z.hom = Y.hom ≫ (G.map f.right.unop).op
⊢ ((𝟭 (Cᵒᵖ ⥤ A)).obj ℱ.val ⋙ coyoneda.obj (op X)).map fV'.op (getSection ℱ hS hx Y ≫ ℱ.val.map f.right) =
    pulledbackFamily ℱ S x Z fV' hV'

Next tactic:
rw [eq] at hV'
@[simp]
theorem getSection_commute {Y Z : StructuredArrow (op U) G.op} (f : Y ⟶ Z) :
    getSection ℱ hS hx Y ≫ ℱ.val.map f.right = getSection ℱ hS hx Z := by
  apply getSection_is_unique
  intro V' fV' hV'
  have eq : Z.hom = Y.hom ≫ (G.map f.right.unop).op := by
    convert f.w
    erw [Category.id_comp]
  rw [eq] at hV'
  

Proof state:
case H
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
Y Z : StructuredArrow (op U) G.op
f : Y ⟶ Z
V' : C
fV' : V' ⟶ Z.right.unop
hV'✝ : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'
hV' : Presieve.functorPullback G (Sieve.pullback (Y.hom ≫ (G.map f.right.unop).op).unop S).arrows fV'
eq : Z.hom = Y.hom ≫ (G.map f.right.unop).op
⊢ ((𝟭 (Cᵒᵖ ⥤ A)).obj ℱ.val ⋙ coyoneda.obj (op X)).map fV'.op (getSection ℱ hS hx Y ≫ ℱ.val.map f.right) =
    pulledbackFamily ℱ S x Z fV' hV'✝

Next tactic:
convert getSection_isAmalgamation ℱ hS hx Y (fV' ≫ f.right.unop) _ using 1
@[simp]
theorem getSection_commute {Y Z : StructuredArrow (op U) G.op} (f : Y ⟶ Z) :
    getSection ℱ hS hx Y ≫ ℱ.val.map f.right = getSection ℱ hS hx Z := by
  apply getSection_is_unique
  intro V' fV' hV'
  have eq : Z.hom = Y.hom ≫ (G.map f.right.unop).op := by
    convert f.w
    erw [Category.id_comp]
  rw [eq] at hV'
  convert getSection_isAmalgamation ℱ hS hx Y (fV' ≫ f.right.unop) _ using 1
  · 

Proof state:
case h.e'_2
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
Y Z : StructuredArrow (op U) G.op
f : Y ⟶ Z
V' : C
fV' : V' ⟶ Z.right.unop
hV'✝ : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'
hV' : Presieve.functorPullback G (Sieve.pullback (Y.hom ≫ (G.map f.right.unop).op).unop S).arrows fV'
eq : Z.hom = Y.hom ≫ (G.map f.right.unop).op
⊢ ((𝟭 (Cᵒᵖ ⥤ A)).obj ℱ.val ⋙ coyoneda.obj (op X)).map fV'.op (getSection ℱ hS hx Y ≫ ℱ.val.map f.right) =
    ((𝟭 (Cᵒᵖ ⥤ A)).obj ℱ.val ⋙ coyoneda.obj (op X)).map (fV' ≫ f.right.unop).op (getSection ℱ hS hx Y)

Next tactic:
aesop_cat
@[simp]
theorem getSection_commute {Y Z : StructuredArrow (op U) G.op} (f : Y ⟶ Z) :
    getSection ℱ hS hx Y ≫ ℱ.val.map f.right = getSection ℱ hS hx Z := by
  apply getSection_is_unique
  intro V' fV' hV'
  have eq : Z.hom = Y.hom ≫ (G.map f.right.unop).op := by
    convert f.w
    erw [Category.id_comp]
  rw [eq] at hV'
  convert getSection_isAmalgamation ℱ hS hx Y (fV' ≫ f.right.unop) _ using 1
  · aesop_cat
  -- porting note: the below proof was mildly rewritten because `simp` changed behaviour
  -- slightly (a rewrite which seemed to work in Lean 3, didn't work in Lean 4 because of
  -- motive is not type correct issues)
  · 

Proof state:
case h.e'_3
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
Y Z : StructuredArrow (op U) G.op
f : Y ⟶ Z
V' : C
fV' : V' ⟶ Z.right.unop
hV'✝ : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'
hV' : Presieve.functorPullback G (Sieve.pullback (Y.hom ≫ (G.map f.right.unop).op).unop S).arrows fV'
eq : Z.hom = Y.hom ≫ (G.map f.right.unop).op
⊢ pulledbackFamily ℱ S x Z fV' hV'✝ = pulledbackFamily ℱ S x Y (fV' ≫ f.right.unop) ?H

Next tactic:
rw [pulledbackFamily_apply, pulledbackFamily_apply]
@[simp]
theorem getSection_commute {Y Z : StructuredArrow (op U) G.op} (f : Y ⟶ Z) :
    getSection ℱ hS hx Y ≫ ℱ.val.map f.right = getSection ℱ hS hx Z := by
  apply getSection_is_unique
  intro V' fV' hV'
  have eq : Z.hom = Y.hom ≫ (G.map f.right.unop).op := by
    convert f.w
    erw [Category.id_comp]
  rw [eq] at hV'
  convert getSection_isAmalgamation ℱ hS hx Y (fV' ≫ f.right.unop) _ using 1
  · aesop_cat
  -- porting note: the below proof was mildly rewritten because `simp` changed behaviour
  -- slightly (a rewrite which seemed to work in Lean 3, didn't work in Lean 4 because of
  -- motive is not type correct issues)
  · rw [pulledbackFamily_apply, pulledbackFamily_apply]
    · 

Proof state:
case h.e'_3
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
Y Z : StructuredArrow (op U) G.op
f : Y ⟶ Z
V' : C
fV' : V' ⟶ Z.right.unop
hV'✝ : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'
hV' : Presieve.functorPullback G (Sieve.pullback (Y.hom ≫ (G.map f.right.unop).op).unop S).arrows fV'
eq : Z.hom = Y.hom ≫ (G.map f.right.unop).op
⊢ x (G.map fV' ≫ Z.hom.unop) hV'✝ ≫ ((Ran.adjunction A G.op).counit.app ℱ.val).app (op V') =
    x (G.map (fV' ≫ f.right.unop) ≫ Y.hom.unop) ?H ≫ ((Ran.adjunction A G.op).counit.app ℱ.val).app (op V')

Next tactic:
congr 2
@[simp]
theorem getSection_commute {Y Z : StructuredArrow (op U) G.op} (f : Y ⟶ Z) :
    getSection ℱ hS hx Y ≫ ℱ.val.map f.right = getSection ℱ hS hx Z := by
  apply getSection_is_unique
  intro V' fV' hV'
  have eq : Z.hom = Y.hom ≫ (G.map f.right.unop).op := by
    convert f.w
    erw [Category.id_comp]
  rw [eq] at hV'
  convert getSection_isAmalgamation ℱ hS hx Y (fV' ≫ f.right.unop) _ using 1
  · aesop_cat
  -- porting note: the below proof was mildly rewritten because `simp` changed behaviour
  -- slightly (a rewrite which seemed to work in Lean 3, didn't work in Lean 4 because of
  -- motive is not type correct issues)
  · rw [pulledbackFamily_apply, pulledbackFamily_apply]
    · congr 2
      

Proof state:
case h.e'_3.e_a.e_f
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
Y Z : StructuredArrow (op U) G.op
f : Y ⟶ Z
V' : C
fV' : V' ⟶ Z.right.unop
hV'✝ : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'
hV' : Presieve.functorPullback G (Sieve.pullback (Y.hom ≫ (G.map f.right.unop).op).unop S).arrows fV'
eq : Z.hom = Y.hom ≫ (G.map f.right.unop).op
⊢ G.map fV' ≫ Z.hom.unop = G.map (fV' ≫ f.right.unop) ≫ Y.hom.unop

Next tactic:
simp [eq]
@[simp]
theorem getSection_commute {Y Z : StructuredArrow (op U) G.op} (f : Y ⟶ Z) :
    getSection ℱ hS hx Y ≫ ℱ.val.map f.right = getSection ℱ hS hx Z := by
  apply getSection_is_unique
  intro V' fV' hV'
  have eq : Z.hom = Y.hom ≫ (G.map f.right.unop).op := by
    convert f.w
    erw [Category.id_comp]
  rw [eq] at hV'
  convert getSection_isAmalgamation ℱ hS hx Y (fV' ≫ f.right.unop) _ using 1
  · aesop_cat
  -- porting note: the below proof was mildly rewritten because `simp` changed behaviour
  -- slightly (a rewrite which seemed to work in Lean 3, didn't work in Lean 4 because of
  -- motive is not type correct issues)
  · rw [pulledbackFamily_apply, pulledbackFamily_apply]
    · congr 2
      simp [eq]
    · 

Proof state:
case H
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
Y Z : StructuredArrow (op U) G.op
f : Y ⟶ Z
V' : C
fV' : V' ⟶ Z.right.unop
hV'✝ : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'
hV' : Presieve.functorPullback G (Sieve.pullback (Y.hom ≫ (G.map f.right.unop).op).unop S).arrows fV'
eq : Z.hom = Y.hom ≫ (G.map f.right.unop).op
⊢ Presieve.functorPullback G (Sieve.pullback Y.hom.unop S).arrows (fV' ≫ f.right.unop)

Next tactic:
change S (G.map _ ≫ Y.hom.unop)
@[simp]
theorem getSection_commute {Y Z : StructuredArrow (op U) G.op} (f : Y ⟶ Z) :
    getSection ℱ hS hx Y ≫ ℱ.val.map f.right = getSection ℱ hS hx Z := by
  apply getSection_is_unique
  intro V' fV' hV'
  have eq : Z.hom = Y.hom ≫ (G.map f.right.unop).op := by
    convert f.w
    erw [Category.id_comp]
  rw [eq] at hV'
  convert getSection_isAmalgamation ℱ hS hx Y (fV' ≫ f.right.unop) _ using 1
  · aesop_cat
  -- porting note: the below proof was mildly rewritten because `simp` changed behaviour
  -- slightly (a rewrite which seemed to work in Lean 3, didn't work in Lean 4 because of
  -- motive is not type correct issues)
  · rw [pulledbackFamily_apply, pulledbackFamily_apply]
    · congr 2
      simp [eq]
    · change S (G.map _ ≫ Y.hom.unop)
      

Proof state:
case H
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
Y Z : StructuredArrow (op U) G.op
f : Y ⟶ Z
V' : C
fV' : V' ⟶ Z.right.unop
hV'✝ : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'
hV' : Presieve.functorPullback G (Sieve.pullback (Y.hom ≫ (G.map f.right.unop).op).unop S).arrows fV'
eq : Z.hom = Y.hom ≫ (G.map f.right.unop).op
⊢ S.arrows (G.map (fV' ≫ f.right.unop) ≫ Y.hom.unop)

Next tactic:
simpa only [Functor.map_comp, Category.assoc] using hV'
/--
A helper lemma for the following two lemmas. Basically stating that if the section `y : X ⟶ 𝒢(V)`
coincides with `x` on `G(V')` for all `G(V') ⊆ V ∈ S`, then `X ⟶ 𝒢(V) ⟶ ℱ(W)` is indeed the
section obtained in `get_sections`. That said, this is littered with some more categorical jargon
in order to be applied in the following lemmas easier.
-/
theorem helper {V} (f : V ⟶ U) (y : X ⟶ ((ran G.op).obj ℱ.val).obj (op V)) (W)
    (H : ∀ {V'} {fV : G.obj V' ⟶ V} (hV), y ≫ ((ran G.op).obj ℱ.val).map fV.op = x (fV ≫ f) hV) :
    y ≫ limit.π (Ran.diagram G.op ℱ.val (op V)) W =
      (gluedLimitCone ℱ hS hx).π.app ((StructuredArrow.map f.op).obj W) := by
  

Proof state:
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
V : D
f : V ⟶ U
y : X ⟶ ((ran G.op).obj ℱ.val).obj (op V)
W : StructuredArrow (op V) G.op
H : ∀ {V' : C} {fV : G.obj V' ⟶ V} (hV : S.arrows (fV ≫ f)), y ≫ ((ran G.op).obj ℱ.val).map fV.op = x (fV ≫ f) hV
⊢ y ≫ limit.π (Ran.diagram G.op ℱ.val (op V)) W = (gluedLimitCone ℱ hS hx).π.app ((StructuredArrow.map f.op).obj W)

Next tactic:
dsimp only [gluedLimitCone_π_app]
/--
A helper lemma for the following two lemmas. Basically stating that if the section `y : X ⟶ 𝒢(V)`
coincides with `x` on `G(V')` for all `G(V') ⊆ V ∈ S`, then `X ⟶ 𝒢(V) ⟶ ℱ(W)` is indeed the
section obtained in `get_sections`. That said, this is littered with some more categorical jargon
in order to be applied in the following lemmas easier.
-/
theorem helper {V} (f : V ⟶ U) (y : X ⟶ ((ran G.op).obj ℱ.val).obj (op V)) (W)
    (H : ∀ {V'} {fV : G.obj V' ⟶ V} (hV), y ≫ ((ran G.op).obj ℱ.val).map fV.op = x (fV ≫ f) hV) :
    y ≫ limit.π (Ran.diagram G.op ℱ.val (op V)) W =
      (gluedLimitCone ℱ hS hx).π.app ((StructuredArrow.map f.op).obj W) := by
  dsimp only [gluedLimitCone_π_app]
  

Proof state:
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
V : D
f : V ⟶ U
y : X ⟶ ((ran G.op).obj ℱ.val).obj (op V)
W : StructuredArrow (op V) G.op
H : ∀ {V' : C} {fV : G.obj V' ⟶ V} (hV : S.arrows (fV ≫ f)), y ≫ ((ran G.op).obj ℱ.val).map fV.op = x (fV ≫ f) hV
⊢ y ≫ limit.π (Ran.diagram G.op ℱ.val (op V)) W = getSection ℱ hS hx ((StructuredArrow.map f.op).obj W)

Next tactic:
apply getSection_is_unique ℱ hS hx ((StructuredArrow.map f.op).obj W)
/--
A helper lemma for the following two lemmas. Basically stating that if the section `y : X ⟶ 𝒢(V)`
coincides with `x` on `G(V')` for all `G(V') ⊆ V ∈ S`, then `X ⟶ 𝒢(V) ⟶ ℱ(W)` is indeed the
section obtained in `get_sections`. That said, this is littered with some more categorical jargon
in order to be applied in the following lemmas easier.
-/
theorem helper {V} (f : V ⟶ U) (y : X ⟶ ((ran G.op).obj ℱ.val).obj (op V)) (W)
    (H : ∀ {V'} {fV : G.obj V' ⟶ V} (hV), y ≫ ((ran G.op).obj ℱ.val).map fV.op = x (fV ≫ f) hV) :
    y ≫ limit.π (Ran.diagram G.op ℱ.val (op V)) W =
      (gluedLimitCone ℱ hS hx).π.app ((StructuredArrow.map f.op).obj W) := by
  dsimp only [gluedLimitCone_π_app]
  apply getSection_is_unique ℱ hS hx ((StructuredArrow.map f.op).obj W)
  

Proof state:
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
V : D
f : V ⟶ U
y : X ⟶ ((ran G.op).obj ℱ.val).obj (op V)
W : StructuredArrow (op V) G.op
H : ∀ {V' : C} {fV : G.obj V' ⟶ V} (hV : S.arrows (fV ≫ f)), y ≫ ((ran G.op).obj ℱ.val).map fV.op = x (fV ≫ f) hV
⊢ IsAmalgamation (pulledbackFamily ℱ S x ((StructuredArrow.map f.op).obj W))
    (y ≫ limit.π (Ran.diagram G.op ℱ.val (op V)) W)

Next tactic:
intro V' fV' hV'
/--
A helper lemma for the following two lemmas. Basically stating that if the section `y : X ⟶ 𝒢(V)`
coincides with `x` on `G(V')` for all `G(V') ⊆ V ∈ S`, then `X ⟶ 𝒢(V) ⟶ ℱ(W)` is indeed the
section obtained in `get_sections`. That said, this is littered with some more categorical jargon
in order to be applied in the following lemmas easier.
-/
theorem helper {V} (f : V ⟶ U) (y : X ⟶ ((ran G.op).obj ℱ.val).obj (op V)) (W)
    (H : ∀ {V'} {fV : G.obj V' ⟶ V} (hV), y ≫ ((ran G.op).obj ℱ.val).map fV.op = x (fV ≫ f) hV) :
    y ≫ limit.π (Ran.diagram G.op ℱ.val (op V)) W =
      (gluedLimitCone ℱ hS hx).π.app ((StructuredArrow.map f.op).obj W) := by
  dsimp only [gluedLimitCone_π_app]
  apply getSection_is_unique ℱ hS hx ((StructuredArrow.map f.op).obj W)
  intro V' fV' hV'
  

Proof state:
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
V : D
f : V ⟶ U
y : X ⟶ ((ran G.op).obj ℱ.val).obj (op V)
W : StructuredArrow (op V) G.op
H : ∀ {V' : C} {fV : G.obj V' ⟶ V} (hV : S.arrows (fV ≫ f)), y ≫ ((ran G.op).obj ℱ.val).map fV.op = x (fV ≫ f) hV
V' : C
fV' : V' ⟶ ((StructuredArrow.map f.op).obj W).right.unop
hV' : Presieve.functorPullback G (Sieve.pullback ((StructuredArrow.map f.op).obj W).hom.unop S).arrows fV'
⊢ ((𝟭 (Cᵒᵖ ⥤ A)).obj ℱ.val ⋙ coyoneda.obj (op X)).map fV'.op (y ≫ limit.π (Ran.diagram G.op ℱ.val (op V)) W) =
    pulledbackFamily ℱ S x ((StructuredArrow.map f.op).obj W) fV' hV'

Next tactic:
dsimp only [Ran.adjunction, Ran.equiv, pulledbackFamily_apply]
/--
A helper lemma for the following two lemmas. Basically stating that if the section `y : X ⟶ 𝒢(V)`
coincides with `x` on `G(V')` for all `G(V') ⊆ V ∈ S`, then `X ⟶ 𝒢(V) ⟶ ℱ(W)` is indeed the
section obtained in `get_sections`. That said, this is littered with some more categorical jargon
in order to be applied in the following lemmas easier.
-/
theorem helper {V} (f : V ⟶ U) (y : X ⟶ ((ran G.op).obj ℱ.val).obj (op V)) (W)
    (H : ∀ {V'} {fV : G.obj V' ⟶ V} (hV), y ≫ ((ran G.op).obj ℱ.val).map fV.op = x (fV ≫ f) hV) :
    y ≫ limit.π (Ran.diagram G.op ℱ.val (op V)) W =
      (gluedLimitCone ℱ hS hx).π.app ((StructuredArrow.map f.op).obj W) := by
  dsimp only [gluedLimitCone_π_app]
  apply getSection_is_unique ℱ hS hx ((StructuredArrow.map f.op).obj W)
  intro V' fV' hV'
  dsimp only [Ran.adjunction, Ran.equiv, pulledbackFamily_apply]
  erw [Adjunction.adjunctionOfEquivRight_counit_app]
  have :
    y ≫ ((ran G.op).obj ℱ.val).map (G.map fV' ≫ W.hom.unop).op =
      x (G.map fV' ≫ W.hom.unop ≫ f) (by 

Proof state:
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
V : D
f : V ⟶ U
y : X ⟶ ((ran G.op).obj ℱ.val).obj (op V)
W : StructuredArrow (op V) G.op
H : ∀ {V' : C} {fV : G.obj V' ⟶ V} (hV : S.arrows (fV ≫ f)), y ≫ ((ran G.op).obj ℱ.val).map fV.op = x (fV ≫ f) hV
V' : C
fV' : V' ⟶ ((StructuredArrow.map f.op).obj W).right.unop
hV' : Presieve.functorPullback G (Sieve.pullback ((StructuredArrow.map f.op).obj W).hom.unop S).arrows fV'
⊢ S.arrows (G.map fV' ≫ W.hom.unop ≫ f)

Next tactic:
simpa only using hV'