l3lab/ntp-mathlib huggingface dataset:
/--
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 simpa only using hV') := 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'
⊢ y ≫ ((ran G.op).obj ℱ.val).map (G.map fV' ≫ W.hom.unop).op = x (G.map fV' ≫ W.hom.unop ≫ f) hV'

Next tactic:
convert H (show S ((G.map fV' ≫ W.hom.unop) ≫ f) by simpa only [Category.assoc] using hV')
      using 2
/--
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 simpa only using hV') := by
    convert H (show S ((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 [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
  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 simpa only using hV') := by
    convert H (show S ((G.map fV' ≫ W.hom.unop) ≫ f) by simpa only [Category.assoc] using hV')
      using 2
    

Proof state:
case h.e'_3.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
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'
⊢ G.map fV' ≫ W.hom.unop ≫ f = (G.map fV' ≫ W.hom.unop) ≫ f

Next tactic:
simp only [Category.assoc]
/--
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 simpa only using hV') := by
    convert H (show S ((G.map fV' ≫ W.hom.unop) ≫ f) by simpa only [Category.assoc] using hV')
      using 2
    simp only [Category.assoc]
  simp only [Quiver.Hom.unop_op, Equiv.symm_symm, StructuredArrow.map_obj_hom, unop_comp,
    Equiv.coe_fn_mk, Functor.comp_map, coyoneda_obj_map, Category.assoc, ← this, op_comp,
    ran_obj_map, NatTrans.id_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
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'
this : y ≫ ((ran G.op).obj ℱ.val).map (G.map fV' ≫ W.hom.unop).op = x (G.map fV' ≫ W.hom.unop ≫ f) hV'
⊢ y ≫ limit.π (Ran.diagram G.op ℱ.val (op V)) W ≫ ((𝟭 (Cᵒᵖ ⥤ A)).obj ℱ.val).map fV'.op =
    y ≫
      limit.pre (Ran.diagram G.op ℱ.val (op V)) (StructuredArrow.map (W.hom.unop.op ≫ (G.map fV').op)) ≫
        𝟙 ((Ran.loc G.op ℱ.val).obj (G.op.obj (op V'))) ≫
          limit.π (Ran.diagram G.op ℱ.val (G.op.obj (op V'))) (StructuredArrow.mk (𝟙 (G.op.obj (op V'))))

Next tactic:
erw [Category.id_comp, limit.pre_π]
/--
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 simpa only using hV') := by
    convert H (show S ((G.map fV' ≫ W.hom.unop) ≫ f) by simpa only [Category.assoc] using hV')
      using 2
    simp only [Category.assoc]
  simp only [Quiver.Hom.unop_op, Equiv.symm_symm, StructuredArrow.map_obj_hom, unop_comp,
    Equiv.coe_fn_mk, Functor.comp_map, coyoneda_obj_map, Category.assoc, ← this, op_comp,
    ran_obj_map, NatTrans.id_app]
  erw [Category.id_comp, limit.pre_π]
  

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'
this : y ≫ ((ran G.op).obj ℱ.val).map (G.map fV' ≫ W.hom.unop).op = x (G.map fV' ≫ W.hom.unop ≫ f) hV'
⊢ y ≫ limit.π (Ran.diagram G.op ℱ.val (op V)) W ≫ ((𝟭 (Cᵒᵖ ⥤ A)).obj ℱ.val).map fV'.op =
    y ≫
      limit.π (Ran.diagram G.op ℱ.val (op V))
        ((StructuredArrow.map (W.hom.unop.op ≫ (G.map fV').op)).obj (StructuredArrow.mk (𝟙 (G.op.obj (op V')))))

Next tactic:
congr
/--
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 simpa only using hV') := by
    convert H (show S ((G.map fV' ≫ W.hom.unop) ≫ f) by simpa only [Category.assoc] using hV')
      using 2
    simp only [Category.assoc]
  simp only [Quiver.Hom.unop_op, Equiv.symm_symm, StructuredArrow.map_obj_hom, unop_comp,
    Equiv.coe_fn_mk, Functor.comp_map, coyoneda_obj_map, Category.assoc, ← this, op_comp,
    ran_obj_map, NatTrans.id_app]
  erw [Category.id_comp, limit.pre_π]
  congr
  

Proof state:
case e_a
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'
this : y ≫ ((ran G.op).obj ℱ.val).map (G.map fV' ≫ W.hom.unop).op = x (G.map fV' ≫ W.hom.unop ≫ f) hV'
⊢ limit.π (Ran.diagram G.op ℱ.val (op V)) W ≫ ((𝟭 (Cᵒᵖ ⥤ A)).obj ℱ.val).map fV'.op =
    limit.π (Ran.diagram G.op ℱ.val (op V))
      ((StructuredArrow.map (W.hom.unop.op ≫ (G.map fV').op)).obj (StructuredArrow.mk (𝟙 (G.op.obj (op V')))))

Next tactic:
convert limit.w (Ran.diagram G.op ℱ.val (op V)) (StructuredArrow.homMk' W fV'.op)
/--
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 simpa only using hV') := by
    convert H (show S ((G.map fV' ≫ W.hom.unop) ≫ f) by simpa only [Category.assoc] using hV')
      using 2
    simp only [Category.assoc]
  simp only [Quiver.Hom.unop_op, Equiv.symm_symm, StructuredArrow.map_obj_hom, unop_comp,
    Equiv.coe_fn_mk, Functor.comp_map, coyoneda_obj_map, Category.assoc, ← this, op_comp,
    ran_obj_map, NatTrans.id_app]
  erw [Category.id_comp, limit.pre_π]
  congr
  convert limit.w (Ran.diagram G.op ℱ.val (op V)) (StructuredArrow.homMk' W fV'.op)
  

Proof state:
case h.e'_3.h.h.e'_7
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'
this : y ≫ ((ran G.op).obj ℱ.val).map (G.map fV' ≫ W.hom.unop).op = x (G.map fV' ≫ W.hom.unop ≫ f) hV'
e_1✝ :
  (((ran G.op).obj ℱ.val).obj (op V) ⟶ ((𝟭 (Cᵒᵖ ⥤ A)).obj ℱ.val).obj (op V')) =
    (limit (Ran.diagram G.op ℱ.val (op V)) ⟶
      (Ran.diagram G.op ℱ.val (op V)).obj (StructuredArrow.mk (W.hom ≫ G.op.map fV'.op)))
⊢ (StructuredArrow.map (W.hom.unop.op ≫ (G.map fV').op)).obj (StructuredArrow.mk (𝟙 (G.op.obj (op V')))) =
    StructuredArrow.mk (W.hom ≫ G.op.map fV'.op)

Next tactic:
rw [StructuredArrow.map_mk]
/--
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 simpa only using hV') := by
    convert H (show S ((G.map fV' ≫ W.hom.unop) ≫ f) by simpa only [Category.assoc] using hV')
      using 2
    simp only [Category.assoc]
  simp only [Quiver.Hom.unop_op, Equiv.symm_symm, StructuredArrow.map_obj_hom, unop_comp,
    Equiv.coe_fn_mk, Functor.comp_map, coyoneda_obj_map, Category.assoc, ← this, op_comp,
    ran_obj_map, NatTrans.id_app]
  erw [Category.id_comp, limit.pre_π]
  congr
  convert limit.w (Ran.diagram G.op ℱ.val (op V)) (StructuredArrow.homMk' W fV'.op)
  rw [StructuredArrow.map_mk]
  

Proof state:
case h.e'_3.h.h.e'_7
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'
this : y ≫ ((ran G.op).obj ℱ.val).map (G.map fV' ≫ W.hom.unop).op = x (G.map fV' ≫ W.hom.unop ≫ f) hV'
e_1✝ :
  (((ran G.op).obj ℱ.val).obj (op V) ⟶ ((𝟭 (Cᵒᵖ ⥤ A)).obj ℱ.val).obj (op V')) =
    (limit (Ran.diagram G.op ℱ.val (op V)) ⟶
      (Ran.diagram G.op ℱ.val (op V)).obj (StructuredArrow.mk (W.hom ≫ G.op.map fV'.op)))
⊢ StructuredArrow.mk ((W.hom.unop.op ≫ (G.map fV').op) ≫ 𝟙 (G.op.obj (op V'))) =
    StructuredArrow.mk (W.hom ≫ G.op.map fV'.op)

Next tactic:
erw [Category.comp_id]
/--
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 simpa only using hV') := by
    convert H (show S ((G.map fV' ≫ W.hom.unop) ≫ f) by simpa only [Category.assoc] using hV')
      using 2
    simp only [Category.assoc]
  simp only [Quiver.Hom.unop_op, Equiv.symm_symm, StructuredArrow.map_obj_hom, unop_comp,
    Equiv.coe_fn_mk, Functor.comp_map, coyoneda_obj_map, Category.assoc, ← this, op_comp,
    ran_obj_map, NatTrans.id_app]
  erw [Category.id_comp, limit.pre_π]
  congr
  convert limit.w (Ran.diagram G.op ℱ.val (op V)) (StructuredArrow.homMk' W fV'.op)
  rw [StructuredArrow.map_mk]
  erw [Category.comp_id]
  

Proof state:
case h.e'_3.h.h.e'_7
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'
this : y ≫ ((ran G.op).obj ℱ.val).map (G.map fV' ≫ W.hom.unop).op = x (G.map fV' ≫ W.hom.unop ≫ f) hV'
e_1✝ :
  (((ran G.op).obj ℱ.val).obj (op V) ⟶ ((𝟭 (Cᵒᵖ ⥤ A)).obj ℱ.val).obj (op V')) =
    (limit (Ran.diagram G.op ℱ.val (op V)) ⟶
      (Ran.diagram G.op ℱ.val (op V)).obj (StructuredArrow.mk (W.hom ≫ G.op.map fV'.op)))
⊢ StructuredArrow.mk (W.hom.unop.op ≫ (G.map fV').op) = StructuredArrow.mk (W.hom ≫ G.op.map fV'.op)

Next tactic:
simp only [Quiver.Hom.unop_op, Functor.op_map, Quiver.Hom.op_unop]
/-- Verify that the `glued_section` is an amalgamation of `x`. -/
theorem gluedSection_isAmalgamation : x.IsAmalgamation (gluedSection ℱ hS hx) := 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
⊢ IsAmalgamation x (gluedSection ℱ hS hx)

Next tactic:
intro V fV hV
/-- Verify that the `glued_section` is an amalgamation of `x`. -/
theorem gluedSection_isAmalgamation : x.IsAmalgamation (gluedSection ℱ hS hx) := by
  intro V fV hV
  -- porting note: next line was `ext W`
  -- Now `ext` can't see that `ran` is defined as a limit.
  -- See https://github.com/leanprover-community/mathlib4/issues/5229
  

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
fV : V ⟶ U
hV : S.arrows fV
⊢ ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)).map fV.op (gluedSection ℱ hS hx) = x fV hV

Next tactic:
refine limit.hom_ext (λ (W : StructuredArrow (op V) G.op) => ?_)