mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Sites.Sheaf

Goal state before:
pp: case mpr.refine_2
C : Type u₁
inst✝ : Category.{v₁, u₁} C
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ Type w
hP : Presieve.IsSheaf J P
X : Type w
Y : C
S : Sieve Y
hS : S ∈ J Y
z : Presieve.FamilyOfElements (P ⋙ coyoneda.obj (op X)) S.arrows
hz : z.Compatible
⊢ (fun t ↦ z.IsAmalgamation t) (↾fun x ↦ ⋯.amalgamate (fun Z f hf ↦ (ConcreteCategory.hom (z f hf)) x) ⋯)

pp: case mpr.refine_3
C : Type u₁
inst✝ : Category.{v₁, u₁} C
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ Type w
hP : Presieve.IsSheaf J P
X : Type w
Y : C
S : Sieve Y
hS : S ∈ J Y
z : Presieve.FamilyOfElements (P ⋙ coyoneda.obj (op X)) S.arrows
hz : z.Compatible
⊢ ∀ (y : (P ⋙ coyoneda.obj (op X)).obj (op Y)),
    (fun t ↦ z.IsAmalgamation t) y → y = ↾fun x ↦ ⋯.amalgamate (fun Z f hf ↦ (ConcreteCategory.hom (z f hf)) x) ⋯

Tactic:
· intro Z f hf
      apply ConcreteCategory.hom_ext
      intro x
      simp only [Functor.comp_obj, Functor.flip_obj_obj, yoneda_obj_obj, Functor.comp_map,
        Functor.flip_obj_map, yoneda_map_app, ConcreteCategory.hom_ofHom, TypeCat.Fun.coe_mk,
        comp_apply]
      apply Presieve.IsSheafFor.valid_glue

Goal state after:
case mpr.refine_3
C : Type u₁
inst✝ : Category.{v₁, u₁} C
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ Type w
hP : Presieve.IsSheaf J P
X : Type w
Y : C
S : Sieve Y
hS : S ∈ J Y
z : Presieve.FamilyOfElements (P ⋙ coyoneda.obj (op X)) S.arrows
hz : z.Compatible
⊢ ∀ (y : (P ⋙ coyoneda.obj (op X)).obj (op Y)),
    (fun t ↦ z.IsAmalgamation t) y → y = ↾fun x ↦ ⋯.amalgamate (fun Z f hf ↦ (ConcreteCategory.hom (z f hf)) x) ⋯
Module: Mathlib.CategoryTheory.Sites.Plus

Goal state before:
pp: case e_a
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝² : Category.{w', w} D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Cᵒᵖ ⥤ D
inst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : unop S ⟶ ⊤ := homOfLE ⋯
I : (unop S).shape.L
ee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯
⊢ Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P I.Y) (op ⊤) =
    colimit.ι (J.diagram P (unop X)) S ≫ (J.plusObj P).map (Cover.Arrow.map I e.op.unop).f.op

Tactic:
← colimit.w _ ee.op,

Goal state after:
case e_a
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝² : Category.{w', w} D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Cᵒᵖ ⥤ D
inst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : unop S ⟶ ⊤ := homOfLE ⋯
I : (unop S).shape.L
ee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯
⊢ Multiequalizer.ι ((unop S).index P) I ≫
      ⊤.toMultiequalizer P ≫
        (J.diagram P I.Y).map ee.op ≫
          colimit.ι (J.diagram P I.Y) (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S))) =
    colimit.ι (J.diagram P (unop X)) S ≫ (J.plusObj P).map (Cover.Arrow.map I e.op.unop).f.op
Module: Mathlib.CategoryTheory.Sites.Subsheaf

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F : Cᵒᵖ ⥤ Type w
G : Subfunctor F
h : Presieve.IsSheaf J F
hG : Presieve.IsSheaf J G.toFunctor
U : Cᵒᵖ
s : F.obj U
hs : s ∈ (sheafify J G).obj U
this : ↑(⋯.amalgamate (G.familyOfElementsOfSection s) ⋯) = s
⊢ ↑(⋯.amalgamate (G.familyOfElementsOfSection s) ⋯) ∈ G.obj U

Tactic:
exact ((hG _ hs).amalgamate _ (G.family_of_elements_compatible s)).2

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Subsheaf

Goal state before:
pp: case hF
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F✝ F'✝ F'' : Cᵒᵖ ⥤ Type w
G G' : Subfunctor F✝
F F' : Sheaf J (Type (max v u))
f : F ⟶ F'
I : Limits.MonoFactorisation f
M : Mono I.m.hom
⊢ Presieve.IsSheaf J F'.obj

pp: case hG'
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F✝ F'✝ F'' : Cᵒᵖ ⥤ Type w
G G' : Subfunctor F✝
F F' : Sheaf J (Type (max v u))
f : F ⟶ F'
I : Limits.MonoFactorisation f
M : Mono I.m.hom
⊢ Presieve.IsSheaf J (Subfunctor.range I.m.hom).toFunctor

Tactic:
· rw [← isSheaf_iff_isSheaf_of_type]
          exact F'.2

Goal state after:
case hG'
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F✝ F'✝ F'' : Cᵒᵖ ⥤ Type w
G G' : Subfunctor F✝
F F' : Sheaf J (Type (max v u))
f : F ⟶ F'
I : Limits.MonoFactorisation f
M : Mono I.m.hom
⊢ Presieve.IsSheaf J (Subfunctor.range I.m.hom).toFunctor
Module: Mathlib.Topology.Category.TopCat.Limits.Products

Goal state before:
pp: case h.right
W X Y Z : TopCat
f : W ⟶ Y
g : X ⟶ Z
x : ↑(Y ⨯ Z)
x₁ : ↑W
hx₁ : (ConcreteCategory.hom f) x₁ = (ConcreteCategory.hom prod.fst) x
x₂ : ↑X
hx₂ : (ConcreteCategory.hom g) x₂ = (ConcreteCategory.hom prod.snd) x
⊢ (ConcreteCategory.hom (limit.π (pair Y Z) { as := WalkingPair.right }))
      ((ConcreteCategory.hom (prod.map f g)) ((ConcreteCategory.hom (W.prodIsoProd X).inv) (x₁, x₂))) =
    (ConcreteCategory.hom (limit.π (pair Y Z) { as := WalkingPair.right })) x

Tactic:
rw [← ConcreteCategory.comp_apply]
      erw [Limits.prod.map_snd]
      rw [ConcreteCategory.comp_apply, TopCat.prodIsoProd_inv_snd_apply]
      exact hx₂

Goal state after:
Module: Mathlib.Topology.Category.TopCat.Limits.Products

Goal state before:
pp: W X Y Z : TopCat
f : W ⟶ X
g : Y ⟶ Z
hf : IsInducing ⇑(ConcreteCategory.hom f)
hg : IsInducing ⇑(ConcreteCategory.hom g)
⊢ induced (⇑(ConcreteCategory.hom prod.fst)) W.str ⊓ induced (⇑(ConcreteCategory.hom prod.snd)) Y.str =
    induced (⇑(ConcreteCategory.hom prod.fst)) (induced (⇑(ConcreteCategory.hom f)) X.str) ⊓
      induced (⇑(ConcreteCategory.hom g) ∘ ⇑(ConcreteCategory.hom prod.snd)) Z.str

Tactic:
← induced_compose (g := g)

Goal state after:
W X Y Z : TopCat
f : W ⟶ X
g : Y ⟶ Z
hf : IsInducing ⇑(ConcreteCategory.hom f)
hg : IsInducing ⇑(ConcreteCategory.hom g)
⊢ induced (⇑(ConcreteCategory.hom prod.fst)) W.str ⊓ induced (⇑(ConcreteCategory.hom prod.snd)) Y.str =
    induced (⇑(ConcreteCategory.hom prod.fst)) (induced (⇑(ConcreteCategory.hom f)) X.str) ⊓
      induced (⇑(ConcreteCategory.hom prod.snd)) (induced (⇑(ConcreteCategory.hom g)) Z.str)
Module: Mathlib.Topology.Category.TopCat.Limits.Pullbacks

Goal state before:
pp: case isOpen_range
W X Y Z S T : TopCat
f₁ : W ⟶ S
f₂ : X ⟶ S
g₁ : Y ⟶ T
g₂ : Z ⟶ T
i₁ : W ⟶ Y
i₂ : X ⟶ Z
H₁ : IsOpenEmbedding ⇑(ConcreteCategory.hom i₁)
H₂ : IsOpenEmbedding ⇑(ConcreteCategory.hom i₂)
i₃ : S ⟶ T
H₃ : Mono i₃
eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁
eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂
⊢ IsOpen (Set.range ⇑(ConcreteCategory.hom (pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂)))

Tactic:
rw [range_pullback_map]

Goal state after:
case isOpen_range
W X Y Z S T : TopCat
f₁ : W ⟶ S
f₂ : X ⟶ S
g₁ : Y ⟶ T
g₂ : Z ⟶ T
i₁ : W ⟶ Y
i₂ : X ⟶ Z
H₁ : IsOpenEmbedding ⇑(ConcreteCategory.hom i₁)
H₂ : IsOpenEmbedding ⇑(ConcreteCategory.hom i₂)
i₃ : S ⟶ T
H₃ : Mono i₃
eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁
eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂
⊢ IsOpen
    (⇑(ConcreteCategory.hom (pullback.fst g₁ g₂)) ⁻¹' Set.range ⇑(ConcreteCategory.hom i₁) ∩
      ⇑(ConcreteCategory.hom (pullback.snd g₁ g₂)) ⁻¹' Set.range ⇑(ConcreteCategory.hom i₂))
Module: Mathlib.CategoryTheory.Limits.MonoCoprod

Goal state before:
pp: case refine_2
C : Type u_1
inst✝ : Category.{v_1, u_1} C
A B : Type u
⊢ Function.Injective ⇑(ConcreteCategory.hom (BinaryCofan.mk (↾Sum.inl) (↾Sum.inr)).inl)

Tactic:
intro a₁

Goal state after:
case refine_2
C : Type u_1
inst✝ : Category.{v_1, u_1} C
A B : Type u
a₁ : (pair A B).obj { as := WalkingPair.left }
⊢ ∀ ⦃a₂ : (fun X ↦ X) ((pair A B).obj { as := WalkingPair.left })⦄,
    (ConcreteCategory.hom (BinaryCofan.mk (↾Sum.inl) (↾Sum.inr)).inl) a₁ =
        (ConcreteCategory.hom (BinaryCofan.mk (↾Sum.inl) (↾Sum.inr)).inl) a₂ →
      a₁ = a₂
Module: Mathlib.CategoryTheory.Limits.MonoCoprod

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
I₁ : Type u_2
I₂ : Type u_3
X : I₁ ⊕ I₂ → C
c : Cofan X
c₁ : Cofan (X ∘ Sum.inl)
c₂ : Cofan (X ∘ Sum.inr)
hc : IsColimit c
hc₁ : IsColimit c₁
hc₂ : IsColimit c₂
inst✝ : MonoCoprod C
inr : c₂.pt ⟶ c.pt
hinr : ∀ (i₂ : I₂), c₂.inj i₂ ≫ inr = c.inj (Sum.inr i₂)
⊢ Mono inr

Tactic:
suffices inr = (binaryCofanSum c c₁ c₂ hc₁ hc₂).inr by
    rw [this]
    exact MonoCoprod.binaryCofan_inr _ (isColimitBinaryCofanSum c c₁ c₂ hc hc₁ hc₂)
  exact Cofan.IsColimit.hom_ext hc₂ _ _ (by simpa using hinr)

Goal state after:
Module: Mathlib.CategoryTheory.Limits.VanKampen

Goal state before:
pp: J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
D : Type u_2
inst✝² : Category.{v_2, u_2} D
inst✝¹ : HasPullbacks D
inst✝ : HasColimitsOfShape J D
F : J ⥤ C ⥤ D
c : Cocone F
hc : ∀ (x : C), IsVanKampenColimit (((evaluation C D).obj x).mapCocone c)
F' : J ⥤ C ⥤ D
c' : Cocone F'
α : F' ⟶ F
f : c'.pt ⟶ c.pt
e : α ≫ c.ι = c'.ι ≫ (const J).map f
hα : NatTrans.Equifibered α
x : C
⊢ whiskerRight α ((evaluation C D).obj x) ≫ (((evaluation C D).obj x).mapCocone c).ι =
    (((evaluation C D).obj x).mapCocone c').ι ≫ (const J).map (((evaluation C D).obj x).map f)

Tactic:
by
        ext y
        dsimp
        exact NatTrans.congr_app (NatTrans.congr_app e y) x

Goal state after:
Module: Mathlib.CategoryTheory.Extensive

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbacksOfInclusions C
T : C
HT : IsTerminal T
c₀ : BinaryCofan T T
hc₀ : IsColimit c₀
H :
  ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ T) (αY : Y' ⟶ T) (f : c'.pt ⟶ c₀.pt),
    αX ≫ c₀.inl = c'.inl ≫ f →
      αY ≫ c₀.inr = c'.inr ≫ f →
        (Nonempty (IsColimit c') ↔ IsPullback c'.inl αX f c₀.inl ∧ IsPullback c'.inr αY f c₀.inr)
X Y : C
c : BinaryCofan X Y
hc : IsColimit c
X' Y' : C
c' : BinaryCofan X' Y'
αX : X' ⟶ X
αY : Y' ⟶ Y
f : c'.pt ⟶ c.pt
hX : αX ≫ c.inl = c'.inl ≫ f
hY : αY ≫ c.inr = c'.inr ≫ f
d : c.pt ⟶ ((Functor.const (Discrete WalkingPair)).obj c₀.pt).obj { as := WalkingPair.left }
hd : c.inl ≫ d = HT.from X ≫ c₀.inl
hd' : c.inr ≫ d = HT.from Y ≫ c₀.inr
⊢ Nonempty (IsColimit c') ↔ IsPullback c'.inl αX f c.inl ∧ IsPullback c'.inr αY f c.inr

Tactic:
H c' (αX ≫ HT.from _) (αY ≫ HT.from _) (f ≫ d) (by rw [← reassoc_of% hX, hd, Category.assoc])
      (by rw [← reassoc_of% hY, hd', Category.assoc])

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasPullbacksOfInclusions C
T : C
HT : IsTerminal T
c₀ : BinaryCofan T T
hc₀ : IsColimit c₀
H :
  ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ T) (αY : Y' ⟶ T) (f : c'.pt ⟶ c₀.pt),
    αX ≫ c₀.inl = c'.inl ≫ f →
      αY ≫ c₀.inr = c'.inr ≫ f →
        (Nonempty (IsColimit c') ↔ IsPullback c'.inl αX f c₀.inl ∧ IsPullback c'.inr αY f c₀.inr)
X Y : C
c : BinaryCofan X Y
hc : IsColimit c
X' Y' : C
c' : BinaryCofan X' Y'
αX : X' ⟶ X
αY : Y' ⟶ Y
f : c'.pt ⟶ c.pt
hX : αX ≫ c.inl = c'.inl ≫ f
hY : αY ≫ c.inr = c'.inr ≫ f
d : c.pt ⟶ ((Functor.const (Discrete WalkingPair)).obj c₀.pt).obj { as := WalkingPair.left }
hd : c.inl ≫ d = HT.from X ≫ c₀.inl
hd' : c.inr ≫ d = HT.from Y ≫ c₀.inr
⊢ IsPullback c'.inl (αX ≫ HT.from X) (f ≫ d) c₀.inl ∧ IsPullback c'.inr (αY ≫ HT.from Y) (f ≫ d) c₀.inr ↔
    IsPullback c'.inl αX f c.inl ∧ IsPullback c'.inr αY f c.inr
Module: Mathlib.CategoryTheory.Limits.FilteredColimitCommutesFiniteLimit

Goal state before:
pp: J : Type u₁
K : Type u₂
inst✝⁴ : SmallCategory J
inst✝³ : Category.{v₂, u₂} K
inst✝² : Small.{v, u₂} K
inst✝¹ : FinCategory J
F : J × K ⥤ Type v
inst✝ : IsFiltered K
x : limit (curry.obj F ⋙ colim)
k : J → K
y : (j : J) → F.obj (j, k j)
e :
  ∀ (j : J),
    (ConcreteCategory.hom (colimit.ι ((curry.obj F).obj j) (k j))) (y j) =
      (ConcreteCategory.hom (limit.π (curry.obj F ⋙ colim) j)) x
k' : K
g : (j : J) → k j ⟶ k'
kf : {j j' : J} → (j ⟶ j') → K
gf hf : {j j' : J} → (f : j ⟶ j') → k' ⟶ kf f
wf :
  ∀ {j j' : J} (f : j ⟶ j'),
    (ConcreteCategory.hom (F.map (𝟙 j' ×ₘ g j' ≫ gf f))) (y j') = (ConcreteCategory.hom (F.map (f ×ₘ g j ≫ hf f))) (y j)
O : Finset K := (Finset.univ.biUnion fun j ↦ Finset.univ.biUnion fun j' ↦ Finset.image kf Finset.univ) ∪ {k'}
kfO : ∀ {j j' : J} (f : j ⟶ j'), kf f ∈ O
⊢ ∃ a, (ConcreteCategory.hom (colimitLimitToLimitColimit F)) a = x

Tactic:
have k'O : k' ∈ O := Finset.mem_union.mpr (Or.inr (Finset.mem_singleton.mpr rfl))

Goal state after:
J : Type u₁
K : Type u₂
inst✝⁴ : SmallCategory J
inst✝³ : Category.{v₂, u₂} K
inst✝² : Small.{v, u₂} K
inst✝¹ : FinCategory J
F : J × K ⥤ Type v
inst✝ : IsFiltered K
x : limit (curry.obj F ⋙ colim)
k : J → K
y : (j : J) → F.obj (j, k j)
e :
  ∀ (j : J),
    (ConcreteCategory.hom (colimit.ι ((curry.obj F).obj j) (k j))) (y j) =
      (ConcreteCategory.hom (limit.π (curry.obj F ⋙ colim) j)) x
k' : K
g : (j : J) → k j ⟶ k'
kf : {j j' : J} → (j ⟶ j') → K
gf hf : {j j' : J} → (f : j ⟶ j') → k' ⟶ kf f
wf :
  ∀ {j j' : J} (f : j ⟶ j'),
    (ConcreteCategory.hom (F.map (𝟙 j' ×ₘ g j' ≫ gf f))) (y j') = (ConcreteCategory.hom (F.map (f ×ₘ g j ≫ hf f))) (y j)
O : Finset K := (Finset.univ.biUnion fun j ↦ Finset.univ.biUnion fun j' ↦ Finset.image kf Finset.univ) ∪ {k'}
kfO : ∀ {j j' : J} (f : j ⟶ j'), kf f ∈ O
k'O : k' ∈ O
⊢ ∃ a, (ConcreteCategory.hom (colimitLimitToLimitColimit F)) a = x
Module: Mathlib.CategoryTheory.Sites.LeftExact

Goal state before:
pp: C : Type u
inst✝¹⁴ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝¹³ : Category.{t, w} D
inst✝¹² : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝¹¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
FD : D → D → Type u_1
CD : D → Type t
inst✝¹⁰ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝⁹ : ConcreteCategory D FD
inst✝⁸ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
inst✝⁷ : (forget D).ReflectsIsomorphisms
inst✝⁶ : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShape (WalkingMulticospan S.shape) (forget D)
K : Type w'
inst✝⁵ : SmallCategory K
inst✝⁴ : FinCategory K
inst✝³ : HasLimitsOfShape K D
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), Small.{t, max u v} (J.Cover X)ᵒᵖ
inst✝ : HasFiniteLimits D
⊢ PreservesFiniteLimits (plusPlusSheaf J D)

Tactic:
apply preservesFiniteLimits_of_preservesFiniteLimitsOfSize.{t}
  intros
  infer_instance

Goal state after:
Module: Mathlib.CategoryTheory.Localization.Adjunction

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₂
inst✝⁵ : Category.{v_3, u_3} D₁
inst✝⁴ : Category.{v_4, u_4} D₂
G : C₁ ⥤ C₂
F : C₂ ⥤ C₁
adj : G ⊣ F
L₁ : C₁ ⥤ D₁
W₁ : MorphismProperty C₁
inst✝³ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
W₂ : MorphismProperty C₂
inst✝² : L₂.IsLocalization W₂
G' : D₁ ⥤ D₂
F' : D₂ ⥤ D₁
inst✝¹ : CatCommSq G L₁ L₂ G'
inst✝ : CatCommSq F L₂ L₁ F'
X₁ : C₁
eq : G.map (adj.unit.app X₁) ≫ adj.counit.app (G.obj X₁) = 𝟙 (G.obj X₁)
⊢ G'.map (L₁.map (adj.unit.app X₁)) ≫
      (CatCommSq.iso G L₁ L₂ G').inv.app (F.obj (G.obj X₁)) ≫
        L₂.map (adj.counit.app (G.obj X₁)) ≫ (𝟭 D₂).map ((CatCommSq.iso G L₁ L₂ G').hom.app X₁) =
    (NatTrans.id (𝟭 D₁ ⋙ G')).app (L₁.obj X₁)

Tactic:
(CatCommSq.iso G L₁ L₂ G').inv.naturality_assoc,

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₂
inst✝⁵ : Category.{v_3, u_3} D₁
inst✝⁴ : Category.{v_4, u_4} D₂
G : C₁ ⥤ C₂
F : C₂ ⥤ C₁
adj : G ⊣ F
L₁ : C₁ ⥤ D₁
W₁ : MorphismProperty C₁
inst✝³ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
W₂ : MorphismProperty C₂
inst✝² : L₂.IsLocalization W₂
G' : D₁ ⥤ D₂
F' : D₂ ⥤ D₁
inst✝¹ : CatCommSq G L₁ L₂ G'
inst✝ : CatCommSq F L₂ L₁ F'
X₁ : C₁
eq : G.map (adj.unit.app X₁) ≫ adj.counit.app (G.obj X₁) = 𝟙 (G.obj X₁)
⊢ (CatCommSq.iso G L₁ L₂ G').inv.app ((𝟭 C₁).obj X₁) ≫
      (G ⋙ L₂).map (adj.unit.app X₁) ≫
        L₂.map (adj.counit.app (G.obj X₁)) ≫ (𝟭 D₂).map ((CatCommSq.iso G L₁ L₂ G').hom.app X₁) =
    (NatTrans.id (𝟭 D₁ ⋙ G')).app (L₁.obj X₁)
Module: Mathlib.CategoryTheory.Limits.VanKampen

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X Y : C
c : BinaryCofan X Y
cofans : (X Y : C) → BinaryCofan X Y
colimits : (X Y : C) → IsColimit (cofans X Y)
cones : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → PullbackCone f g
limits : {X Y Z : C} → (f : X ⟶ Z) → (g : Y ⟶ Z) → IsLimit (cones f g)
h₁ :
  ∀ {X' Y' : C} (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : (cofans X' Y').pt ⟶ c.pt),
    αX ≫ c.inl = (cofans X' Y').inl ≫ f →
      αY ≫ c.inr = (cofans X' Y').inr ≫ f →
        IsPullback (cofans X' Y').inl αX f c.inl ∧ IsPullback (cofans X' Y').inr αY f c.inr
h₂ : {Z : C} → (f : Z ⟶ c.pt) → IsColimit (BinaryCofan.mk (cones f c.inl).fst (cones f c.inr).fst)
X' Y' : C
c' : BinaryCofan X' Y'
αX : X' ⟶ X
αY : Y' ⟶ Y
f : c'.pt ⟶ c.pt
hX : αX ≫ c.inl = c'.inl ≫ f
hY : αY ≫ c.inr = c'.inr ≫ f
h : IsColimit c'
e : c'.pt ≅ (cofans X' Y').pt := h.coconePointUniqueUpToIso (colimits X' Y')
⊢ αY ≫ c.inr = (cofans X' Y').inr ≫ e.inv ≫ f

Tactic:
by simp [e, hY]

Goal state after:
Module: Mathlib.CategoryTheory.Sites.PreservesSheafification

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u_1
B : Type u_2
inst✝² : Category.{v_1, u_1} A
inst✝¹ : Category.{v_2, u_2} B
F : A ⥤ B
G₁ : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A
adj₁ : G₁ ⊣ sheafToPresheaf J A
G₂ : (Cᵒᵖ ⥤ B) ⥤ Sheaf J B
adj₂ : G₂ ⊣ sheafToPresheaf J B
inst✝ : J.HasSheafCompose F
P Q : Cᵒᵖ ⥤ A
f : P ⟶ Q
⊢ ((whiskeringRight Cᵒᵖ A B).obj F ⋙ G₂).map f ≫
      (adj₂.homEquiv (((whiskeringRight Cᵒᵖ A B).obj F).obj Q) ((G₁ ⋙ sheafCompose J F).obj Q)).symm
        (whiskerRight (adj₁.unit.app Q) F) =
    (adj₂.homEquiv (((whiskeringRight Cᵒᵖ A B).obj F).obj P) ((G₁ ⋙ sheafCompose J F).obj P)).symm
        (whiskerRight (adj₁.unit.app P) F) ≫
      (G₁ ⋙ sheafCompose J F).map f

Tactic:
dsimp
    erw [← adj₂.homEquiv_naturality_left_symm,
      ← adj₂.homEquiv_naturality_right_symm]
    congr 1
    ext X
    have := NatTrans.congr_app (adj₁.unit.naturality f) X
    dsimp at this ⊢
    grind

Goal state after:
Module: Mathlib.CategoryTheory.Sites.PreservesSheafification

Goal state before:
pp: C : Type u
inst✝¹⁸ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u_1
B : Type u_2
inst✝¹⁷ : Category.{v_1, u_1} A
inst✝¹⁶ : Category.{v_2, u_2} B
F✝ : A ⥤ B
D : Type u_3
E : Type u_4
inst✝¹⁵ : Category.{v_3, u_3} D
inst✝¹⁴ : Category.{v_4, u_4} E
F : D ⥤ E
inst✝¹³ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝¹² : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝¹¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹⁰ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁹ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝⁸ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
FD : D → D → Type u_5
CD : D → Type u_6
FE : E → E → Type u_7
CE : E → Type u_8
inst✝⁷ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝⁶ : (X Y : E) → FunLike (FE X Y) (CE X) (CE Y)
instCCD : ConcreteCategory D FD
instCCE : ConcreteCategory E FE
inst✝⁵ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
inst✝⁴ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget E)
inst✝³ : PreservesLimitsOfSize.{max v u, max v u, v_3, u_6, u_3, u_6 + 1} (forget D)
inst✝² : PreservesLimitsOfSize.{max v u, max v u, v_4, u_8, u_4, u_8 + 1} (forget E)
inst✝¹ : (forget D).ReflectsIsomorphisms
inst✝ : (forget E).ReflectsIsomorphisms
P : Cᵒᵖ ⥤ D
⊢ IsIso ((sheafComposeNatTrans J F (plusPlusAdjunction J D) (plusPlusAdjunction J E)).app P)

Tactic:
← isIso_iff_of_reflects_iso _ (sheafToPresheaf J E),

Goal state after:
C : Type u
inst✝¹⁸ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u_1
B : Type u_2
inst✝¹⁷ : Category.{v_1, u_1} A
inst✝¹⁶ : Category.{v_2, u_2} B
F✝ : A ⥤ B
D : Type u_3
E : Type u_4
inst✝¹⁵ : Category.{v_3, u_3} D
inst✝¹⁴ : Category.{v_4, u_4} E
F : D ⥤ E
inst✝¹³ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝¹² : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝¹¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹⁰ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁹ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝⁸ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
FD : D → D → Type u_5
CD : D → Type u_6
FE : E → E → Type u_7
CE : E → Type u_8
inst✝⁷ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝⁶ : (X Y : E) → FunLike (FE X Y) (CE X) (CE Y)
instCCD : ConcreteCategory D FD
instCCE : ConcreteCategory E FE
inst✝⁵ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
inst✝⁴ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget E)
inst✝³ : PreservesLimitsOfSize.{max v u, max v u, v_3, u_6, u_3, u_6 + 1} (forget D)
inst✝² : PreservesLimitsOfSize.{max v u, max v u, v_4, u_8, u_4, u_8 + 1} (forget E)
inst✝¹ : (forget D).ReflectsIsomorphisms
inst✝ : (forget E).ReflectsIsomorphisms
P : Cᵒᵖ ⥤ D
⊢ IsIso ((sheafToPresheaf J E).map ((sheafComposeNatTrans J F (plusPlusAdjunction J D) (plusPlusAdjunction J E)).app P))
Module: Mathlib.Algebra.Category.ModuleCat.Sheaf

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R✝ : Sheaf J RingCat
R : Cᵒᵖ ⥤ RingCat
M₁ M₂ : PresheafOfModules R
f : M₁ ⟶ M₂
N : PresheafOfModules R
hN : Presheaf.IsSheaf J N.presheaf
inst✝² : J.WEqualsLocallyBijective AddCommGrpCat
inst✝¹ : IsLocallySurjective J f
inst✝ : IsLocallyInjective J f
ψ : M₁ ⟶ N
φ : (toPresheaf R).obj M₂ ⟶ N.presheaf
hφ :
  ∀ (Z : Cᵒᵖ) (x : ↑(M₁.obj Z)),
    (ConcreteCategory.hom (φ.app Z)) ((ConcreteCategory.hom (f.app Z)) x) = (ConcreteCategory.hom (ψ.app Z)) x
X : Cᵒᵖ
r : ↑(R.obj X)
y : ↑(M₂.obj X)
Y : C
p : Y ⟶ Opposite.unop X
x : ↑(M₁.obj (Opposite.op Y))
hx : (ConcreteCategory.hom (f.app (Opposite.op Y))) x = (ConcreteCategory.hom (M₂.map p.op)) y
hφ' :
  ∀ (z : ↑(M₂.obj X)),
    (ConcreteCategory.hom (φ.app (Opposite.op Y))) ((ConcreteCategory.hom (M₂.map p.op)) z) =
      (ConcreteCategory.hom (N.map p.op)) ((ConcreteCategory.hom (φ.app (Opposite.op (Opposite.unop X)))) z)
⊢ (ConcreteCategory.hom (φ.app (Opposite.op Y)))
      ((ConcreteCategory.hom (R.map p.op)) r • (ConcreteCategory.hom (f.app (Opposite.op Y))) x) =
    (ConcreteCategory.hom (N.map p.op)) (r • (ConcreteCategory.hom (φ.app X)) y)

Tactic:
← (f.app _).hom.map_smul,

Goal state after:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R✝ : Sheaf J RingCat
R : Cᵒᵖ ⥤ RingCat
M₁ M₂ : PresheafOfModules R
f : M₁ ⟶ M₂
N : PresheafOfModules R
hN : Presheaf.IsSheaf J N.presheaf
inst✝² : J.WEqualsLocallyBijective AddCommGrpCat
inst✝¹ : IsLocallySurjective J f
inst✝ : IsLocallyInjective J f
ψ : M₁ ⟶ N
φ : (toPresheaf R).obj M₂ ⟶ N.presheaf
hφ :
  ∀ (Z : Cᵒᵖ) (x : ↑(M₁.obj Z)),
    (ConcreteCategory.hom (φ.app Z)) ((ConcreteCategory.hom (f.app Z)) x) = (ConcreteCategory.hom (ψ.app Z)) x
X : Cᵒᵖ
r : ↑(R.obj X)
y : ↑(M₂.obj X)
Y : C
p : Y ⟶ Opposite.unop X
x : ↑(M₁.obj (Opposite.op Y))
hx : (ConcreteCategory.hom (f.app (Opposite.op Y))) x = (ConcreteCategory.hom (M₂.map p.op)) y
hφ' :
  ∀ (z : ↑(M₂.obj X)),
    (ConcreteCategory.hom (φ.app (Opposite.op Y))) ((ConcreteCategory.hom (M₂.map p.op)) z) =
      (ConcreteCategory.hom (N.map p.op)) ((ConcreteCategory.hom (φ.app (Opposite.op (Opposite.unop X)))) z)
⊢ (ConcreteCategory.hom (φ.app (Opposite.op Y)))
      ((ModuleCat.Hom.hom (f.app (Opposite.op Y))) ((ConcreteCategory.hom (R.map p.op)) r • x)) =
    (ConcreteCategory.hom (N.map p.op)) (r • (ConcreteCategory.hom (φ.app X)) y)