l3lab/ntp-mathlib huggingface dataset:
theorem exists_rep {X : C} {P : Cᵒᵖ ⥤ D} (x : (J.plusObj P).obj (op X)) :
    ∃ (S : J.Cover X) (y : Meq P S), x = mk y := by
  obtain ⟨S, y, h⟩ := Concrete.colimit_exists_rep (J.diagram P X) x
  

Proof state:
case intro.intro
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
X : C
P : Cᵒᵖ ⥤ D
x : (forget D).obj ((plusObj J P).obj (op X))
S : (Cover J X)ᵒᵖ
y : (forget D).obj ((diagram J P X).obj S)
h : (colimit.ι (diagram J P X) S) y = x
⊢ ∃ S y, x = mk y

Next tactic:
use S.unop, Meq.equiv _ _ y
theorem exists_rep {X : C} {P : Cᵒᵖ ⥤ D} (x : (J.plusObj P).obj (op X)) :
    ∃ (S : J.Cover X) (y : Meq P S), x = mk y := by
  obtain ⟨S, y, h⟩ := Concrete.colimit_exists_rep (J.diagram P X) x
  use S.unop, Meq.equiv _ _ y
  

Proof state:
case h
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
X : C
P : Cᵒᵖ ⥤ D
x : (forget D).obj ((plusObj J P).obj (op X))
S : (Cover J X)ᵒᵖ
y : (forget D).obj ((diagram J P X).obj S)
h : (colimit.ι (diagram J P X) S) y = x
⊢ x = mk ((Meq.equiv P S.unop) y)

Next tactic:
rw [← h]
theorem exists_rep {X : C} {P : Cᵒᵖ ⥤ D} (x : (J.plusObj P).obj (op X)) :
    ∃ (S : J.Cover X) (y : Meq P S), x = mk y := by
  obtain ⟨S, y, h⟩ := Concrete.colimit_exists_rep (J.diagram P X) x
  use S.unop, Meq.equiv _ _ y
  rw [← h]
  

Proof state:
case h
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
X : C
P : Cᵒᵖ ⥤ D
x : (forget D).obj ((plusObj J P).obj (op X))
S : (Cover J X)ᵒᵖ
y : (forget D).obj ((diagram J P X).obj S)
h : (colimit.ι (diagram J P X) S) y = x
⊢ (colimit.ι (diagram J P X) S) y = mk ((Meq.equiv P S.unop) y)

Next tactic:
dsimp [mk]
theorem exists_rep {X : C} {P : Cᵒᵖ ⥤ D} (x : (J.plusObj P).obj (op X)) :
    ∃ (S : J.Cover X) (y : Meq P S), x = mk y := by
  obtain ⟨S, y, h⟩ := Concrete.colimit_exists_rep (J.diagram P X) x
  use S.unop, Meq.equiv _ _ y
  rw [← h]
  dsimp [mk]
  

Proof state:
case h
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
X : C
P : Cᵒᵖ ⥤ D
x : (forget D).obj ((plusObj J P).obj (op X))
S : (Cover J X)ᵒᵖ
y : (forget D).obj ((diagram J P X).obj S)
h : (colimit.ι (diagram J P X) S) y = x
⊢ (colimit.ι (diagram J P X) S) y = (colimit.ι (diagram J P X) S) ((Meq.equiv P S.unop).symm ((Meq.equiv P S.unop) y))

Next tactic:
simp
theorem eq_mk_iff_exists {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.Cover X} (x : Meq P S) (y : Meq P T) :
    mk x = mk y ↔ ∃ (W : J.Cover X) (h1 : W ⟶ S) (h2 : W ⟶ T), x.refine h1 = y.refine h2 := by
  

Proof state:
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
X : C
P : Cᵒᵖ ⥤ D
S T : Cover J X
x : Meq P S
y : Meq P T
⊢ mk x = mk y ↔ ∃ W h1 h2, Meq.refine x h1 = Meq.refine y h2

Next tactic:
constructor
theorem eq_mk_iff_exists {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.Cover X} (x : Meq P S) (y : Meq P T) :
    mk x = mk y ↔ ∃ (W : J.Cover X) (h1 : W ⟶ S) (h2 : W ⟶ T), x.refine h1 = y.refine h2 := by
  constructor
  · 

Proof state:
case mp
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
X : C
P : Cᵒᵖ ⥤ D
S T : Cover J X
x : Meq P S
y : Meq P T
⊢ mk x = mk y → ∃ W h1 h2, Meq.refine x h1 = Meq.refine y h2

Next tactic:
intro h
theorem eq_mk_iff_exists {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.Cover X} (x : Meq P S) (y : Meq P T) :
    mk x = mk y ↔ ∃ (W : J.Cover X) (h1 : W ⟶ S) (h2 : W ⟶ T), x.refine h1 = y.refine h2 := by
  constructor
  · intro h
    

Proof state:
case mp
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
X : C
P : Cᵒᵖ ⥤ D
S T : Cover J X
x : Meq P S
y : Meq P T
h : mk x = mk y
⊢ ∃ W h1 h2, Meq.refine x h1 = Meq.refine y h2

Next tactic:
obtain ⟨W, h1, h2, hh⟩ := Concrete.colimit_exists_of_rep_eq _ _ _ h
theorem eq_mk_iff_exists {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.Cover X} (x : Meq P S) (y : Meq P T) :
    mk x = mk y ↔ ∃ (W : J.Cover X) (h1 : W ⟶ S) (h2 : W ⟶ T), x.refine h1 = y.refine h2 := by
  constructor
  · intro h
    obtain ⟨W, h1, h2, hh⟩ := Concrete.colimit_exists_of_rep_eq _ _ _ h
    

Proof state:
case mp.intro.intro.intro
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
X : C
P : Cᵒᵖ ⥤ D
S T : Cover J X
x : Meq P S
y : Meq P T
h : mk x = mk y
W : (Cover J X)ᵒᵖ
h1 : op S ⟶ W
h2 : op T ⟶ W
hh : ((diagram J P X).map h1) ((Meq.equiv P S).symm x) = ((diagram J P X).map h2) ((Meq.equiv P T).symm y)
⊢ ∃ W h1 h2, Meq.refine x h1 = Meq.refine y h2

Next tactic:
use W.unop, h1.unop, h2.unop
theorem eq_mk_iff_exists {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.Cover X} (x : Meq P S) (y : Meq P T) :
    mk x = mk y ↔ ∃ (W : J.Cover X) (h1 : W ⟶ S) (h2 : W ⟶ T), x.refine h1 = y.refine h2 := by
  constructor
  · intro h
    obtain ⟨W, h1, h2, hh⟩ := Concrete.colimit_exists_of_rep_eq _ _ _ h
    use W.unop, h1.unop, h2.unop
    

Proof state:
case h
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
X : C
P : Cᵒᵖ ⥤ D
S T : Cover J X
x : Meq P S
y : Meq P T
h : mk x = mk y
W : (Cover J X)ᵒᵖ
h1 : op S ⟶ W
h2 : op T ⟶ W
hh : ((diagram J P X).map h1) ((Meq.equiv P S).symm x) = ((diagram J P X).map h2) ((Meq.equiv P T).symm y)
⊢ Meq.refine x h1.unop = Meq.refine y h2.unop

Next tactic:
ext I
theorem eq_mk_iff_exists {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.Cover X} (x : Meq P S) (y : Meq P T) :
    mk x = mk y ↔ ∃ (W : J.Cover X) (h1 : W ⟶ S) (h2 : W ⟶ T), x.refine h1 = y.refine h2 := by
  constructor
  · intro h
    obtain ⟨W, h1, h2, hh⟩ := Concrete.colimit_exists_of_rep_eq _ _ _ h
    use W.unop, h1.unop, h2.unop
    ext I
    

Proof state:
case h.h
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
X : C
P : Cᵒᵖ ⥤ D
S T : Cover J X
x : Meq P S
y : Meq P T
h : mk x = mk y
W : (Cover J X)ᵒᵖ
h1 : op S ⟶ W
h2 : op T ⟶ W
hh : ((diagram J P X).map h1) ((Meq.equiv P S).symm x) = ((diagram J P X).map h2) ((Meq.equiv P T).symm y)
I : Cover.Arrow W.unop
⊢ ↑(Meq.refine x h1.unop) I = ↑(Meq.refine y h2.unop) I

Next tactic:
apply_fun Multiequalizer.ι (W.unop.index P) I at hh
theorem eq_mk_iff_exists {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.Cover X} (x : Meq P S) (y : Meq P T) :
    mk x = mk y ↔ ∃ (W : J.Cover X) (h1 : W ⟶ S) (h2 : W ⟶ T), x.refine h1 = y.refine h2 := by
  constructor
  · intro h
    obtain ⟨W, h1, h2, hh⟩ := Concrete.colimit_exists_of_rep_eq _ _ _ h
    use W.unop, h1.unop, h2.unop
    ext I
    apply_fun Multiequalizer.ι (W.unop.index P) I at hh
    

Proof state:
case h.h
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
X : C
P : Cᵒᵖ ⥤ D
S T : Cover J X
x : Meq P S
y : Meq P T
h : mk x = mk y
W : (Cover J X)ᵒᵖ
h1 : op S ⟶ W
h2 : op T ⟶ W
I : Cover.Arrow W.unop
hh :
  (Multiequalizer.ι (Cover.index W.unop P) I) (((diagram J P X).map h1) ((Meq.equiv P S).symm x)) =
    (Multiequalizer.ι (Cover.index W.unop P) I) (((diagram J P X).map h2) ((Meq.equiv P T).symm y))
⊢ ↑(Meq.refine x h1.unop) I = ↑(Meq.refine y h2.unop) I

Next tactic:
convert hh
theorem eq_mk_iff_exists {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.Cover X} (x : Meq P S) (y : Meq P T) :
    mk x = mk y ↔ ∃ (W : J.Cover X) (h1 : W ⟶ S) (h2 : W ⟶ T), x.refine h1 = y.refine h2 := by
  constructor
  · intro h
    obtain ⟨W, h1, h2, hh⟩ := Concrete.colimit_exists_of_rep_eq _ _ _ h
    use W.unop, h1.unop, h2.unop
    ext I
    apply_fun Multiequalizer.ι (W.unop.index P) I at hh
    convert hh
    

Proof state:
case h.e'_2.h
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
X : C
P : Cᵒᵖ ⥤ D
S T : Cover J X
x : Meq P S
y : Meq P T
h : mk x = mk y
W : (Cover J X)ᵒᵖ
h1 : op S ⟶ W
h2 : op T ⟶ W
I : Cover.Arrow W.unop
hh :
  (Multiequalizer.ι (Cover.index W.unop P) I) (((diagram J P X).map h1) ((Meq.equiv P S).symm x)) =
    (Multiequalizer.ι (Cover.index W.unop P) I) (((diagram J P X).map h2) ((Meq.equiv P T).symm y))
e_1✝ : (forget D).obj (P.obj (op I.Y)) = (forget D).obj (MulticospanIndex.left (Cover.index W.unop P) I)
⊢ ↑(Meq.refine x h1.unop) I =
    (Multiequalizer.ι (Cover.index W.unop P) I) (((diagram J P X).map h1) ((Meq.equiv P S).symm x))
case h.e'_3.h
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
X : C
P : Cᵒᵖ ⥤ D
S T : Cover J X
x : Meq P S
y : Meq P T
h : mk x = mk y
W : (Cover J X)ᵒᵖ
h1 : op S ⟶ W
h2 : op T ⟶ W
I : Cover.Arrow W.unop
hh :
  (Multiequalizer.ι (Cover.index W.unop P) I) (((diagram J P X).map h1) ((Meq.equiv P S).symm x)) =
    (Multiequalizer.ι (Cover.index W.unop P) I) (((diagram J P X).map h2) ((Meq.equiv P T).symm y))
e_1✝ : (forget D).obj (P.obj (op I.Y)) = (forget D).obj (MulticospanIndex.left (Cover.index W.unop P) I)
⊢ ↑(Meq.refine y h2.unop) I =
    (Multiequalizer.ι (Cover.index W.unop P) I) (((diagram J P X).map h2) ((Meq.equiv P T).symm y))

Next tactic:
all_goals
      dsimp [diagram]
      erw [← comp_apply, Multiequalizer.lift_ι, Meq.equiv_symm_eq_apply]
      cases I; rfl
theorem eq_mk_iff_exists {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.Cover X} (x : Meq P S) (y : Meq P T) :
    mk x = mk y ↔ ∃ (W : J.Cover X) (h1 : W ⟶ S) (h2 : W ⟶ T), x.refine h1 = y.refine h2 := by
  constructor
  · intro h
    obtain ⟨W, h1, h2, hh⟩ := Concrete.colimit_exists_of_rep_eq _ _ _ h
    use W.unop, h1.unop, h2.unop
    ext I
    apply_fun Multiequalizer.ι (W.unop.index P) I at hh
    convert hh
    all_goals
      

Proof state:
case h.e'_2.h
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
X : C
P : Cᵒᵖ ⥤ D
S T : Cover J X
x : Meq P S
y : Meq P T
h : mk x = mk y
W : (Cover J X)ᵒᵖ
h1 : op S ⟶ W
h2 : op T ⟶ W
I : Cover.Arrow W.unop
hh :
  (Multiequalizer.ι (Cover.index W.unop P) I) (((diagram J P X).map h1) ((Meq.equiv P S).symm x)) =
    (Multiequalizer.ι (Cover.index W.unop P) I) (((diagram J P X).map h2) ((Meq.equiv P T).symm y))
e_1✝ : (forget D).obj (P.obj (op I.Y)) = (forget D).obj (MulticospanIndex.left (Cover.index W.unop P) I)
⊢ ↑(Meq.refine x h1.unop) I =
    (Multiequalizer.ι (Cover.index W.unop P) I) (((diagram J P X).map h1) ((Meq.equiv P S).symm x))

Next tactic:
dsimp [diagram]
theorem eq_mk_iff_exists {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.Cover X} (x : Meq P S) (y : Meq P T) :
    mk x = mk y ↔ ∃ (W : J.Cover X) (h1 : W ⟶ S) (h2 : W ⟶ T), x.refine h1 = y.refine h2 := by
  constructor
  · intro h
    obtain ⟨W, h1, h2, hh⟩ := Concrete.colimit_exists_of_rep_eq _ _ _ h
    use W.unop, h1.unop, h2.unop
    ext I
    apply_fun Multiequalizer.ι (W.unop.index P) I at hh
    convert hh
    all_goals
      dsimp [diagram]
      

Proof state:
case h.e'_2.h
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
X : C
P : Cᵒᵖ ⥤ D
S T : Cover J X
x : Meq P S
y : Meq P T
h : mk x = mk y
W : (Cover J X)ᵒᵖ
h1 : op S ⟶ W
h2 : op T ⟶ W
I : Cover.Arrow W.unop
hh :
  (Multiequalizer.ι (Cover.index W.unop P) I) (((diagram J P X).map h1) ((Meq.equiv P S).symm x)) =
    (Multiequalizer.ι (Cover.index W.unop P) I) (((diagram J P X).map h2) ((Meq.equiv P T).symm y))
e_1✝ : (forget D).obj (P.obj (op I.Y)) = (forget D).obj (MulticospanIndex.left (Cover.index W.unop P) I)
⊢ ↑x { Y := I.Y, f := I.f, hf := (_ : ((fun a => ↑a) S).arrows I.f) } =
    (Multiequalizer.ι (Cover.index W.unop P) I)
      ((Multiequalizer.lift (Cover.index W.unop P) (multiequalizer (Cover.index S P))
          (fun I => Multiequalizer.ι (Cover.index S P) (Cover.Arrow.map I h1.unop))
          (_ :
            ∀ (I : (Cover.index W.unop P).R),
              Multiequalizer.ι (Cover.index (op S).unop P)
                    (MulticospanIndex.fstTo (Cover.index (op S).unop P) (Cover.Relation.map I h1.unop)) ≫
                  MulticospanIndex.fst (Cover.index (op S).unop P) (Cover.Relation.map I h1.unop) =
                Multiequalizer.ι (Cover.index (op S).unop P)
                    (MulticospanIndex.sndTo (Cover.index (op S).unop P) (Cover.Relation.map I h1.unop)) ≫
                  MulticospanIndex.snd (Cover.index (op S).unop P) (Cover.Relation.map I h1.unop)))
        ((Meq.equiv P S).symm x))

Next tactic:
erw [← comp_apply, Multiequalizer.lift_ι, Meq.equiv_symm_eq_apply]
theorem eq_mk_iff_exists {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.Cover X} (x : Meq P S) (y : Meq P T) :
    mk x = mk y ↔ ∃ (W : J.Cover X) (h1 : W ⟶ S) (h2 : W ⟶ T), x.refine h1 = y.refine h2 := by
  constructor
  · intro h
    obtain ⟨W, h1, h2, hh⟩ := Concrete.colimit_exists_of_rep_eq _ _ _ h
    use W.unop, h1.unop, h2.unop
    ext I
    apply_fun Multiequalizer.ι (W.unop.index P) I at hh
    convert hh
    all_goals
      dsimp [diagram]
      erw [← comp_apply, Multiequalizer.lift_ι, Meq.equiv_symm_eq_apply]
      

Proof state:
case h.e'_2.h
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
X : C
P : Cᵒᵖ ⥤ D
S T : Cover J X
x : Meq P S
y : Meq P T
h : mk x = mk y
W : (Cover J X)ᵒᵖ
h1 : op S ⟶ W
h2 : op T ⟶ W
I : Cover.Arrow W.unop
hh :
  (Multiequalizer.ι (Cover.index W.unop P) I) (((diagram J P X).map h1) ((Meq.equiv P S).symm x)) =
    (Multiequalizer.ι (Cover.index W.unop P) I) (((diagram J P X).map h2) ((Meq.equiv P T).symm y))
e_1✝ : (forget D).obj (P.obj (op I.Y)) = (forget D).obj (MulticospanIndex.left (Cover.index W.unop P) I)
⊢ ↑x { Y := I.Y, f := I.f, hf := (_ : ((fun a => ↑a) S).arrows I.f) } = ↑x (Cover.Arrow.map I h1.unop)

Next tactic:
cases I
theorem eq_mk_iff_exists {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.Cover X} (x : Meq P S) (y : Meq P T) :
    mk x = mk y ↔ ∃ (W : J.Cover X) (h1 : W ⟶ S) (h2 : W ⟶ T), x.refine h1 = y.refine h2 := by
  constructor
  · intro h
    obtain ⟨W, h1, h2, hh⟩ := Concrete.colimit_exists_of_rep_eq _ _ _ h
    use W.unop, h1.unop, h2.unop
    ext I
    apply_fun Multiequalizer.ι (W.unop.index P) I at hh
    convert hh
    all_goals
      dsimp [diagram]
      erw [← comp_apply, Multiequalizer.lift_ι, Meq.equiv_symm_eq_apply]
      cases I; 

Proof state:
case h.e'_2.h.mk
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
X : C
P : Cᵒᵖ ⥤ D
S T : Cover J X
x : Meq P S
y : Meq P T
h : mk x = mk y
W : (Cover J X)ᵒᵖ
h1 : op S ⟶ W
h2 : op T ⟶ W
Y✝ : C
f✝ : Y✝ ⟶ X
hf✝ : (Cover.sieve W.unop).arrows f✝
hh :
  (Multiequalizer.ι (Cover.index W.unop P) { Y := Y✝, f := f✝, hf := hf✝ })
      (((diagram J P X).map h1) ((Meq.equiv P S).symm x)) =
    (Multiequalizer.ι (Cover.index W.unop P) { Y := Y✝, f := f✝, hf := hf✝ })
      (((diagram J P X).map h2) ((Meq.equiv P T).symm y))
e_1✝ :
  (forget D).obj (P.obj (op { Y := Y✝, f := f✝, hf := hf✝ }.Y)) =
    (forget D).obj (MulticospanIndex.left (Cover.index W.unop P) { Y := Y✝, f := f✝, hf := hf✝ })
⊢ ↑x
      { Y := { Y := Y✝, f := f✝, hf := hf✝ }.Y, f := { Y := Y✝, f := f✝, hf := hf✝ }.f,
        hf := (_ : ((fun a => ↑a) S).arrows { Y := Y✝, f := f✝, hf := hf✝ }.f) } =
    ↑x (Cover.Arrow.map { Y := Y✝, f := f✝, hf := hf✝ } h1.unop)

Next tactic:
rfl
theorem eq_mk_iff_exists {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.Cover X} (x : Meq P S) (y : Meq P T) :
    mk x = mk y ↔ ∃ (W : J.Cover X) (h1 : W ⟶ S) (h2 : W ⟶ T), x.refine h1 = y.refine h2 := by
  constructor
  · intro h
    obtain ⟨W, h1, h2, hh⟩ := Concrete.colimit_exists_of_rep_eq _ _ _ h
    use W.unop, h1.unop, h2.unop
    ext I
    apply_fun Multiequalizer.ι (W.unop.index P) I at hh
    convert hh
    all_goals
      

Proof state:
case h.e'_3.h
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
X : C
P : Cᵒᵖ ⥤ D
S T : Cover J X
x : Meq P S
y : Meq P T
h : mk x = mk y
W : (Cover J X)ᵒᵖ
h1 : op S ⟶ W
h2 : op T ⟶ W
I : Cover.Arrow W.unop
hh :
  (Multiequalizer.ι (Cover.index W.unop P) I) (((diagram J P X).map h1) ((Meq.equiv P S).symm x)) =
    (Multiequalizer.ι (Cover.index W.unop P) I) (((diagram J P X).map h2) ((Meq.equiv P T).symm y))
e_1✝ : (forget D).obj (P.obj (op I.Y)) = (forget D).obj (MulticospanIndex.left (Cover.index W.unop P) I)
⊢ ↑(Meq.refine y h2.unop) I =
    (Multiequalizer.ι (Cover.index W.unop P) I) (((diagram J P X).map h2) ((Meq.equiv P T).symm y))

Next tactic:
dsimp [diagram]
theorem eq_mk_iff_exists {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.Cover X} (x : Meq P S) (y : Meq P T) :
    mk x = mk y ↔ ∃ (W : J.Cover X) (h1 : W ⟶ S) (h2 : W ⟶ T), x.refine h1 = y.refine h2 := by
  constructor
  · intro h
    obtain ⟨W, h1, h2, hh⟩ := Concrete.colimit_exists_of_rep_eq _ _ _ h
    use W.unop, h1.unop, h2.unop
    ext I
    apply_fun Multiequalizer.ι (W.unop.index P) I at hh
    convert hh
    all_goals
      dsimp [diagram]
      

Proof state:
case h.e'_3.h
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
X : C
P : Cᵒᵖ ⥤ D
S T : Cover J X
x : Meq P S
y : Meq P T
h : mk x = mk y
W : (Cover J X)ᵒᵖ
h1 : op S ⟶ W
h2 : op T ⟶ W
I : Cover.Arrow W.unop
hh :
  (Multiequalizer.ι (Cover.index W.unop P) I) (((diagram J P X).map h1) ((Meq.equiv P S).symm x)) =
    (Multiequalizer.ι (Cover.index W.unop P) I) (((diagram J P X).map h2) ((Meq.equiv P T).symm y))
e_1✝ : (forget D).obj (P.obj (op I.Y)) = (forget D).obj (MulticospanIndex.left (Cover.index W.unop P) I)
⊢ ↑y { Y := I.Y, f := I.f, hf := (_ : ((fun a => ↑a) T).arrows I.f) } =
    (Multiequalizer.ι (Cover.index W.unop P) I)
      ((Multiequalizer.lift (Cover.index W.unop P) (multiequalizer (Cover.index T P))
          (fun I => Multiequalizer.ι (Cover.index T P) (Cover.Arrow.map I h2.unop))
          (_ :
            ∀ (I : (Cover.index W.unop P).R),
              Multiequalizer.ι (Cover.index (op T).unop P)
                    (MulticospanIndex.fstTo (Cover.index (op T).unop P) (Cover.Relation.map I h2.unop)) ≫
                  MulticospanIndex.fst (Cover.index (op T).unop P) (Cover.Relation.map I h2.unop) =
                Multiequalizer.ι (Cover.index (op T).unop P)
                    (MulticospanIndex.sndTo (Cover.index (op T).unop P) (Cover.Relation.map I h2.unop)) ≫
                  MulticospanIndex.snd (Cover.index (op T).unop P) (Cover.Relation.map I h2.unop)))
        ((Meq.equiv P T).symm y))

Next tactic:
erw [← comp_apply, Multiequalizer.lift_ι, Meq.equiv_symm_eq_apply]