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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
X B : C
π : X ⟶ B
inst✝ : EffectiveEpi π
c : PullbackCone π π
hc : IsLimit c
this : HasPullback π π
hP :
  ∀
    (b :
      (fun X ↦ X)
        ↑{x |
            (ConcreteCategory.hom (P.map (pullback.fst π π).op)) x =
              (ConcreteCategory.hom (P.map (pullback.snd π π).op)) x}),
    ∃! a, (ConcreteCategory.hom (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯)) a = b
b : P.obj (op X)
hb : (ConcreteCategory.hom (P.map c.fst.op)) b = (ConcreteCategory.hom (P.map c.snd.op)) b
h₁ : ((pullbackIsPullback π π).conePointUniqueUpToIso hc).hom ≫ c.fst = pullback.fst π π
⊢ (ConcreteCategory.hom (P.map (c.fst.op ≫ ((pullbackIsPullback π π).conePointUniqueUpToIso hc).hom.op))) b =
    (ConcreteCategory.hom (P.map (pullback.snd π π).op)) b

Tactic:
Functor.map_comp,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
X B : C
π : X ⟶ B
inst✝ : EffectiveEpi π
c : PullbackCone π π
hc : IsLimit c
this : HasPullback π π
hP :
  ∀
    (b :
      (fun X ↦ X)
        ↑{x |
            (ConcreteCategory.hom (P.map (pullback.fst π π).op)) x =
              (ConcreteCategory.hom (P.map (pullback.snd π π).op)) x}),
    ∃! a, (ConcreteCategory.hom (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯)) a = b
b : P.obj (op X)
hb : (ConcreteCategory.hom (P.map c.fst.op)) b = (ConcreteCategory.hom (P.map c.snd.op)) b
h₁ : ((pullbackIsPullback π π).conePointUniqueUpToIso hc).hom ≫ c.fst = pullback.fst π π
⊢ (ConcreteCategory.hom (P.map c.fst.op ≫ P.map ((pullbackIsPullback π π).conePointUniqueUpToIso hc).hom.op)) b =
    (ConcreteCategory.hom (P.map (pullback.snd π π).op)) b
Module: Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preregular C
W X w✝ Y : C
f : Y ⟶ X
hf : EffectiveEpi f
hS : ∃ f_1, ((ofArrows (fun x ↦ Y) fun x ↦ f) = ofArrows (fun x ↦ w✝) fun x ↦ f_1) ∧ EffectiveEpi f_1
this : (ofArrows (fun x ↦ Y) fun x ↦ f).regular
h_colim : IsColimit (Sieve.generateSingleton f).arrows.cocone
⊢ IsSheafFor (yoneda.obj W) (ofArrows (fun x ↦ Y) fun x ↦ f)

Tactic:
rw [← Sieve.generateSingleton_eq, ← Presieve.ofArrows_pUnit] at h_colim

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preregular C
W X w✝ Y : C
f : Y ⟶ X
hf : EffectiveEpi f
hS : ∃ f_1, ((ofArrows (fun x ↦ Y) fun x ↦ f) = ofArrows (fun x ↦ w✝) fun x ↦ f_1) ∧ EffectiveEpi f_1
this : (ofArrows (fun x ↦ Y) fun x ↦ f).regular
h_colim : IsColimit (Sieve.generate (ofArrows (fun x ↦ Y) fun x ↦ f)).arrows.cocone
⊢ IsSheafFor (yoneda.obj W) (ofArrows (fun x ↦ Y) fun x ↦ f)
Module: Mathlib.CategoryTheory.Sites.Hypercover.Homotopy

Goal state before:
pp: case refine_2
C : Type u
inst✝¹ : Category.{v, u} C
S : C
E : PreOneHypercover S
F : PreOneHypercover S
A : Type u_1
inst✝ : Category.{v_1, u_1} A
f : E.Hom F
g : F.Hom E
hgf : Homotopy (g.comp f) (Hom.id F)
G : Cᵒᵖ ⥤ A
hE : IsLimit (E.multifork G)
t : Multifork (F.multicospanIndex G)
i : F.multicospanShape.L
h1 : hgf.a i ≫ F.p₁ (hgf.H i) = g.h₀ i ≫ f.h₀ (g.s₀ i)
h2 : t.ι (f.s₀ (g.s₀ i)) ≫ G.map (F.p₁ (hgf.H i)).op = t.ι i ≫ G.map (F.p₂ (hgf.H i)).op
⊢ Multifork.IsLimit.lift hE (fun a ↦ t.ι (f.s₀ a) ≫ G.map (f.h₀ a).op) ⋯ ≫ G.map ((E.f (g.s₀ i)).op ≫ (g.h₀ i).op) =
    t.ι i

Tactic:
Functor.map_comp,

Goal state after:
case refine_2
C : Type u
inst✝¹ : Category.{v, u} C
S : C
E : PreOneHypercover S
F : PreOneHypercover S
A : Type u_1
inst✝ : Category.{v_1, u_1} A
f : E.Hom F
g : F.Hom E
hgf : Homotopy (g.comp f) (Hom.id F)
G : Cᵒᵖ ⥤ A
hE : IsLimit (E.multifork G)
t : Multifork (F.multicospanIndex G)
i : F.multicospanShape.L
h1 : hgf.a i ≫ F.p₁ (hgf.H i) = g.h₀ i ≫ f.h₀ (g.s₀ i)
h2 : t.ι (f.s₀ (g.s₀ i)) ≫ G.map (F.p₁ (hgf.H i)).op = t.ι i ≫ G.map (F.p₂ (hgf.H i)).op
⊢ Multifork.IsLimit.lift hE (fun a ↦ t.ι (f.s₀ a) ≫ G.map (f.h₀ a).op) ⋯ ≫ G.map (E.f (g.s₀ i)).op ≫ G.map (g.h₀ i).op =
    t.ι i
Module: Mathlib.CategoryTheory.Sites.Hypercover.Homotopy

Goal state before:
pp: case refine_2
C : Type u
inst✝¹ : Category.{v, u} C
S : C
E : PreOneHypercover S
F : PreOneHypercover S
A : Type u_1
inst✝ : Category.{v_1, u_1} A
f : E.Hom F
g : F.Hom E
hgf : Homotopy (g.comp f) (Hom.id F)
G : Cᵒᵖ ⥤ A
hE : IsLimit (E.multifork G)
t : Multifork (F.multicospanIndex G)
i : F.multicospanShape.L
h1 : hgf.a i ≫ F.p₁ (hgf.H i) = g.h₀ i ≫ f.h₀ (g.s₀ i)
h2 : t.ι (f.s₀ (g.s₀ i)) ≫ G.map (F.p₁ (hgf.H i)).op = t.ι i ≫ G.map (F.p₂ (hgf.H i)).op
⊢ t.ι (f.s₀ (g.s₀ i)) ≫ G.map ((F.p₁ (hgf.H i)).op ≫ (hgf.a i).op) = t.ι i

Tactic:
Functor.map_comp,

Goal state after:
case refine_2
C : Type u
inst✝¹ : Category.{v, u} C
S : C
E : PreOneHypercover S
F : PreOneHypercover S
A : Type u_1
inst✝ : Category.{v_1, u_1} A
f : E.Hom F
g : F.Hom E
hgf : Homotopy (g.comp f) (Hom.id F)
G : Cᵒᵖ ⥤ A
hE : IsLimit (E.multifork G)
t : Multifork (F.multicospanIndex G)
i : F.multicospanShape.L
h1 : hgf.a i ≫ F.p₁ (hgf.H i) = g.h₀ i ≫ f.h₀ (g.s₀ i)
h2 : t.ι (f.s₀ (g.s₀ i)) ≫ G.map (F.p₁ (hgf.H i)).op = t.ι i ≫ G.map (F.p₂ (hgf.H i)).op
⊢ t.ι (f.s₀ (g.s₀ i)) ≫ G.map (F.p₁ (hgf.H i)).op ≫ G.map (hgf.a i).op = t.ι i
Module: Mathlib.CategoryTheory.Sites.Hypercover.Homotopy

Goal state before:
pp: case refine_3
C : Type u
inst✝¹ : Category.{v, u} C
S : C
E : PreOneHypercover S
F : PreOneHypercover S
A : Type u_1
inst✝ : Category.{v_1, u_1} A
f : E.Hom F
g : F.Hom E
hgf : Homotopy (g.comp f) (Hom.id F)
G : Cᵒᵖ ⥤ A
hE : IsLimit (E.multifork G)
t : Multifork (F.multicospanIndex G)
m : t.pt ⟶ (F.multifork G).pt
hm : ∀ (i : F.multicospanShape.L), m ≫ (F.multifork G).ι i = t.ι i
i : E.multicospanShape.L
⊢ m ≫ G.map ((F.f (f.s₀ i)).op ≫ (f.h₀ i).op) = t.ι (f.s₀ i) ≫ G.map (f.h₀ i).op

Tactic:
Functor.map_comp,

Goal state after:
case refine_3
C : Type u
inst✝¹ : Category.{v, u} C
S : C
E : PreOneHypercover S
F : PreOneHypercover S
A : Type u_1
inst✝ : Category.{v_1, u_1} A
f : E.Hom F
g : F.Hom E
hgf : Homotopy (g.comp f) (Hom.id F)
G : Cᵒᵖ ⥤ A
hE : IsLimit (E.multifork G)
t : Multifork (F.multicospanIndex G)
m : t.pt ⟶ (F.multifork G).pt
hm : ∀ (i : F.multicospanShape.L), m ≫ (F.multifork G).ι i = t.ι i
i : E.multicospanShape.L
⊢ m ≫ G.map (F.f (f.s₀ i)).op ≫ G.map (f.h₀ i).op = t.ι (f.s₀ i) ≫ G.map (f.h₀ i).op
Module: Mathlib.CategoryTheory.Sites.Hypercover.Subcanonical

Goal state before:
pp: case refine_1
C : Type u
inst✝³ : Category.{v, u} C
J : Precoverage C
inst✝² : J.toGrothendieck.Subcanonical
inst✝¹ : Limits.HasPullbacks C
inst✝ : J.IsStableUnderBaseChange
X Y : C
f : X ⟶ Y
𝒰 : J.ZeroHypercover Y
H : ∀ (i : 𝒰.I₀), IsIso (pullback.snd f (𝒰.f i))
⊢ Y ⟶ X

Tactic:
refine 𝒰.glueMorphisms (fun i ↦ inv (pullback.snd f (𝒰.f i)) ≫ pullback.fst _ _) fun i j ↦ ?_

Goal state after:
case refine_1
C : Type u
inst✝³ : Category.{v, u} C
J : Precoverage C
inst✝² : J.toGrothendieck.Subcanonical
inst✝¹ : Limits.HasPullbacks C
inst✝ : J.IsStableUnderBaseChange
X Y : C
f : X ⟶ Y
𝒰 : J.ZeroHypercover Y
H : ∀ (i : 𝒰.I₀), IsIso (pullback.snd f (𝒰.f i))
i j : 𝒰.I₀
⊢ pullback.fst (𝒰.f i) (𝒰.f j) ≫ inv (pullback.snd f (𝒰.f i)) ≫ pullback.fst f (𝒰.f i) =
    pullback.snd (𝒰.f i) (𝒰.f j) ≫ inv (pullback.snd f (𝒰.f j)) ≫ pullback.fst f (𝒰.f j)
Module: Mathlib.CategoryTheory.Sites.Hypercover.Subcanonical

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : Precoverage C
inst✝² : J.toGrothendieck.Subcanonical
inst✝¹ : Limits.HasPullbacks C
inst✝ : J.IsStableUnderBaseChange
P X Y Z : C
fst : P ⟶ X
snd : P ⟶ Y
f : X ⟶ Z
g : Y ⟶ Z
𝒰 : J.ZeroHypercover X
H : ∀ (i : 𝒰.I₀), IsPullback (pullback.snd fst (𝒰.f i)) (pullback.fst fst (𝒰.f i) ≫ snd) (𝒰.f i ≫ f) g
h : fst ≫ f = snd ≫ g
i : (𝒰.pullbackCoverOfLeft f g).I₀
m : pullback (𝒰.f i ≫ f) g ⟶ pullback f g := pullback.map (𝒰.f i ≫ f) g f g (𝒰.f i) (𝟙 Y) (𝟙 Z) ⋯ ⋯
this : IsPullback (pullback.fst (𝒰.f i ≫ f) g) m (𝒰.f i) (pullback.fst f g)
⊢ IsPullback (pullback.fst fst (𝒰.f i)) (pullback.lift (pullback.snd fst (𝒰.f i)) (pullback.fst fst (𝒰.f i) ≫ snd) ⋯)
    (pullback.lift fst snd h) m

Tactic:
rw [← IsPullback.paste_vert_iff this.flip (by ext <;> simp [m, pullback.condition])]

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
J : Precoverage C
inst✝² : J.toGrothendieck.Subcanonical
inst✝¹ : Limits.HasPullbacks C
inst✝ : J.IsStableUnderBaseChange
P X Y Z : C
fst : P ⟶ X
snd : P ⟶ Y
f : X ⟶ Z
g : Y ⟶ Z
𝒰 : J.ZeroHypercover X
H : ∀ (i : 𝒰.I₀), IsPullback (pullback.snd fst (𝒰.f i)) (pullback.fst fst (𝒰.f i) ≫ snd) (𝒰.f i ≫ f) g
h : fst ≫ f = snd ≫ g
i : (𝒰.pullbackCoverOfLeft f g).I₀
m : pullback (𝒰.f i ≫ f) g ⟶ pullback f g := pullback.map (𝒰.f i ≫ f) g f g (𝒰.f i) (𝟙 Y) (𝟙 Z) ⋯ ⋯
this : IsPullback (pullback.fst (𝒰.f i ≫ f) g) m (𝒰.f i) (pullback.fst f g)
⊢ IsPullback (pullback.fst fst (𝒰.f i))
    (pullback.lift (pullback.snd fst (𝒰.f i)) (pullback.fst fst (𝒰.f i) ≫ snd) ⋯ ≫ pullback.fst (𝒰.f i ≫ f) g)
    (pullback.lift fst snd h ≫ pullback.fst f g) (𝒰.f i)
Module: Mathlib.CategoryTheory.Sites.SheafHom

Goal state before:
pp: case h
C : Type u
inst✝¹ : Category.{v, u} C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X : C
S : Sieve X
hG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)
x : Presieve.FamilyOfElements (presheafHom F G) S.arrows
Y : C
hx : x.Compatible
g : Y ⟶ X
c : Cone ((Sieve.pullback g S).arrows.diagram.op ⋙ G) := ⋯
⊢ ∀ {Z : C} (p : Z ⟶ Y) (hp : S.arrows (p ≫ g)),
    (hG g).lift c ≫ G.map p.op = F.map p.op ≫ (x (p ≫ g) hp).app (op (Over.mk (𝟙 Z)))

Tactic:
intro Z

Goal state after:
case h
C : Type u
inst✝¹ : Category.{v, u} C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X : C
S : Sieve X
hG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)
x : Presieve.FamilyOfElements (presheafHom F G) S.arrows
Y : C
hx : x.Compatible
g : Y ⟶ X
c : Cone ((Sieve.pullback g S).arrows.diagram.op ⋙ G) :=
  { pt := F.obj (op Y),
    π :=
      {
        app := fun x_1 ↦
          match x_1 with
          | op { obj := Z, property := hZ } =>
            F.map Z.hom.op ≫ (x (Z.hom ≫ g) hZ).app (op (Over.mk (𝟙 (unop (op Z.left))))),
        naturality := ⋯ } }
Z : C
⊢ ∀ (p : Z ⟶ Y) (hp : S.arrows (p ≫ g)),
    (hG g).lift c ≫ G.map p.op = F.map p.op ≫ (x (p ≫ g) hp).app (op (Over.mk (𝟙 Z)))
Module: Mathlib.CategoryTheory.Sites.Monoidal

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
A : Type u₃
inst✝³ : Category.{v₃, u₃} A
inst✝² : MonoidalCategory A
inst✝¹ : MonoidalClosed A
M : A
F G : Cᵒᵖ ⥤ A
inst✝ : HasFunctorEnrichedHom A F G
X : C
g : (presheafHom (F ⊗ (Functor.const Cᵒᵖ).obj M) G).obj (op X)
j j' : Under (op X)
φ : j ⟶ j'
⊢ curry (g.app (op (Over.mk j.hom.unop))) ≫ (ihom (F.obj j.right)).map (G.map (Under.Hom.right φ)) =
    curry (g.app (op (Over.mk j'.hom.unop))) ≫ (pre (F.map (Under.Hom.right φ))).app (G.obj j'.right)

Tactic:
curry_pre_app,

Goal state after:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
A : Type u₃
inst✝³ : Category.{v₃, u₃} A
inst✝² : MonoidalCategory A
inst✝¹ : MonoidalClosed A
M : A
F G : Cᵒᵖ ⥤ A
inst✝ : HasFunctorEnrichedHom A F G
X : C
g : (presheafHom (F ⊗ (Functor.const Cᵒᵖ).obj M) G).obj (op X)
j j' : Under (op X)
φ : j ⟶ j'
⊢ curry (g.app (op (Over.mk j.hom.unop))) ≫ (ihom (F.obj j.right)).map (G.map (Under.Hom.right φ)) =
    curry (F.map (Under.Hom.right φ) ▷ M ≫ g.app (op (Over.mk j'.hom.unop)))
Module: Mathlib.CategoryTheory.Sites.SheafHom

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X : C
S : Sieve X
hG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)
x : Presieve.FamilyOfElements (presheafHom F G) S.arrows
hx : x.Compatible
Y₁ Y₂ : Over X
φ : Y₂ ⟶ Y₁
Z : Over Y₂.left
hZ : (Sieve.pullback Y₂.hom S).arrows Z.hom
⊢ F.map ((Over.Hom.left φ).op ≫ Z.hom.op) ≫ (x (Z.hom ≫ Y₂.hom) hZ).app (op (Over.mk (𝟙 Z.left))) =
    F.map ((Over.Hom.left φ).op ≫ Z.hom.op) ≫ (x ((Z.hom ≫ Over.Hom.left φ) ≫ Y₁.hom) ⋯).app (op (Over.mk (𝟙 Z.left)))

Tactic:
congr 3

Goal state after:
case e_a
C : Type u
inst✝¹ : Category.{v, u} C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X : C
S : Sieve X
hG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)
x : Presieve.FamilyOfElements (presheafHom F G) S.arrows
hx : x.Compatible
Y₁ Y₂ : Over X
φ : Y₂ ⟶ Y₁
Z : Over Y₂.left
hZ : (Sieve.pullback Y₂.hom S).arrows Z.hom
⊢ Z.hom ≫ Y₂.hom = (Z.hom ≫ Over.Hom.left φ) ≫ Y₁.hom
Module: Mathlib.CategoryTheory.Sites.Descent.DescentData

Goal state before:
pp: case refine_2
C : Type u
inst✝ : Category.{v, u} C
F : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat
S : C
R : Sieve S
S₀ : C
M N : ↑(F.obj { as := op S₀ })
a : S ⟶ S₀
h :
  Presieve.IsSheafFor (F.presheafHom M N)
    (Sieve.pullback (Over.isoMk (Iso.refl ((Over.map a).obj (Over.mk (𝟙 S))).left) ⋯).inv
        (Sieve.functorPushforward (Over.map a) ((Sieve.overEquiv (Over.mk (𝟙 S))).symm R))).arrows
⊢ Sieve.pullback (Over.isoMk (Iso.refl ((Over.map a).obj (Over.mk (𝟙 S))).left) ⋯).inv
      (Sieve.functorPushforward (Over.map a) ((Sieve.overEquiv (Over.mk (𝟙 S))).symm R)) ≤
    (Sieve.overEquiv (Over.mk a)).symm R

Tactic:
· rintro X b ⟨Y, c, d, h, fac⟩
    replace fac := (Over.forget _).congr_map fac
    dsimp at fac
    rw [Category.comp_id] at fac
    change R.arrows b.left
    simpa [fac] using R.downward_closed h d.left

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Generators

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : HasZeroObject C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : Preadditive C
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
P Q : ObjectProperty C
inst✝¹ : Q.IsStableUnderRetracts
inst✝ : Q.IsTriangulated
h : P ≤ Q
n : ℕ
⊢ (P.shiftClosure ℤ).binaryProductsClosure.retractClosure.extensionProductIter n ≤ Q

Tactic:
apply extensionProductIter_le_of_isTriangulatedClosed₂

Goal state after:
C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : HasZeroObject C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : Preadditive C
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
P Q : ObjectProperty C
inst✝¹ : Q.IsStableUnderRetracts
inst✝ : Q.IsTriangulated
h : P ≤ Q
n : ℕ
⊢ (P.shiftClosure ℤ).binaryProductsClosure.retractClosure ≤ Q
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : HasShift C ℤ
inst✝¹ : HasShift D ℤ
F : C ⥤ D
inst✝ : F.CommShift ℤ
X : Cᵒᵖ
n : ℤ
⊢ (commShiftIso F n).hom.app (unop ((shiftFunctor Cᵒᵖ n).obj X)) ≫
      (shiftFunctor D n).map (F.map ((shiftFunctorOpIso C n (-n) ⋯).inv.app X).unop) ≫
        (shiftFunctor D n).map ((commShiftIso F (-n)).hom.app (unop X)) ≫
          (shiftFunctorCompIsoId D (-n) n ⋯).hom.app (F.obj (unop X)) =
    (commShiftIso F n).hom.app (unop ((shiftFunctor Cᵒᵖ n).obj X)) ≫
      (shiftFunctor D n).map (F.map ((shiftFunctorOpIso C n (-n) ⋯).inv.app X).unop) ≫
        (shiftFunctor D n).map ((commShiftIso F (-n)).hom.app (unop X)) ≫
          (shiftFunctor D n).map ((shiftFunctorOpIso D n (-n) ⋯).hom.app (op (F.obj (unop X)))).unop ≫
            (shiftFunctor D n).map ((shiftFunctorOpIso D n (-n) ⋯).inv.app (op (F.obj (unop X)))).unop ≫
              (shiftFunctorCompIsoId D (-n) n ⋯).hom.app (F.obj (unop X))

Tactic:
congr 3

Goal state after:
case e_a
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : HasShift C ℤ
inst✝¹ : HasShift D ℤ
F : C ⥤ D
inst✝ : F.CommShift ℤ
X : Cᵒᵖ
n : ℤ
⊢ (shiftFunctorCompIsoId D (-n) n ⋯).hom.app (F.obj (unop X)) =
    (shiftFunctor D n).map ((shiftFunctorOpIso D n (-n) ⋯).hom.app (op (F.obj (unop X)))).unop ≫
      (shiftFunctor D n).map ((shiftFunctorOpIso D n (-n) ⋯).inv.app (op (F.obj (unop X)))).unop ≫
        (shiftFunctorCompIsoId D (-n) n ⋯).hom.app (F.obj (unop X))
Module: Mathlib.CategoryTheory.Triangulated.Adjunction

Goal state before:
pp: case refine_2
C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : HasZeroObject C
inst✝¹² : HasZeroObject D
inst✝¹¹ : Preadditive C
inst✝¹⁰ : Preadditive D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : HasShift D ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : Pretriangulated D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝³ : F.CommShift ℤ
inst✝² : G.CommShift ℤ
inst✝¹ : adj.CommShift ℤ
inst✝ : F.IsTriangulated
T : Triangle D
hT : T ∈ distinguishedTriangles
this✝¹ : G.Additive
Z : C
f : G.obj T.obj₂ ⟶ Z
g : Z ⟶ (shiftFunctor C 1).obj (G.obj T.obj₁)
mem : Triangle.mk (G.map T.mor₁) f g ∈ distinguishedTriangles
h : F.obj Z ⟶ T.obj₃
h₁ : F.map f ≫ h = adj.counit.app T.obj₂ ≫ T.mor₂
h₂ :
  F.map g ≫ (Functor.commShiftIso F 1).hom.app (G.obj T.obj₁) ≫ (shiftFunctor D 1).map (adj.counit.app T.obj₁) =
    h ≫ T.mor₃
h₁' : f ≫ adj.unit.app Z ≫ G.map h = G.map T.mor₂
h₂' : g ≫ (Functor.commShiftIso G 1).inv.app T.obj₁ = (adj.homEquiv Z T.obj₃) h ≫ G.map T.mor₃
this✝ : Mono ((adj.homEquiv Z T.obj₃) h)
this : IsIso ((adj.homEquiv Z T.obj₃) h)
⊢ (adj.homEquiv (G.obj T.obj₂) T.obj₃).symm (f ≫ (adj.homEquiv Z T.obj₃) h) =
    (adj.homEquiv (G.obj T.obj₂) T.obj₃).symm (𝟙 (G.obj T.obj₂) ≫ G.map T.mor₂)

Tactic:
simp only [homEquiv_unit, homEquiv_counit, Functor.map_comp, assoc,
        counit_naturality, left_triangle_components_assoc, h₁, id_comp]

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Goal state before:
pp: C : Type u_1
A : Type u_2
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasZeroObject C
inst✝⁷ : Preadditive C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) f₂ f₃ ∈ distinguishedTriangles
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
inst✝² : ι.Full
inst✝¹ : Preadditive A
inst✝ : ι.Faithful
B : A
x₁ : B ⟶ X₁
hx₁ : x₁ ≫ f₁ = 0
k' : (shiftFunctor C 1).obj (ι.obj B) ⟶ (shiftFunctor C 1).obj (ι.obj K)
hk' : (shiftFunctor C 1).map (ι.map x₁) = k' ≫ α ≫ f₃
⊢ (ι ⋙ shiftFunctor C 1).map ((ι ⋙ shiftFunctor C 1).preimage k' ≫ ιK f₃ α) = (ι ⋙ shiftFunctor C 1).map x₁

Tactic:
Functor.map_comp,

Goal state after:
C : Type u_1
A : Type u_2
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasZeroObject C
inst✝⁷ : Preadditive C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Category.{v_2, u_2} A
ι : A ⥤ C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ (shiftFunctor C 1).obj (ι.obj X₁)
hT : Triangle.mk (ι.map f₁) f₂ f₃ ∈ distinguishedTriangles
K Q : A
α : (shiftFunctor C 1).obj (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ : ι.obj Q ⟶ (shiftFunctor C 1).obj ((shiftFunctor C 1).obj (ι.obj K))
hT' : Triangle.mk α β γ ∈ distinguishedTriangles
inst✝² : ι.Full
inst✝¹ : Preadditive A
inst✝ : ι.Faithful
B : A
x₁ : B ⟶ X₁
hx₁ : x₁ ≫ f₁ = 0
k' : (shiftFunctor C 1).obj (ι.obj B) ⟶ (shiftFunctor C 1).obj (ι.obj K)
hk' : (shiftFunctor C 1).map (ι.map x₁) = k' ≫ α ≫ f₃
⊢ (ι ⋙ shiftFunctor C 1).map ((ι ⋙ shiftFunctor C 1).preimage k') ≫ (ι ⋙ shiftFunctor C 1).map (ιK f₃ α) =
    (ι ⋙ shiftFunctor C 1).map x₁
Module: Mathlib.CategoryTheory.Triangulated.Adjunction

Goal state before:
pp: case refine_3
C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : HasZeroObject C
inst✝¹² : HasZeroObject D
inst✝¹¹ : Preadditive C
inst✝¹⁰ : Preadditive D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : HasShift D ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : Pretriangulated D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝³ : F.CommShift ℤ
inst✝² : G.CommShift ℤ
inst✝¹ : adj.CommShift ℤ
inst✝ : F.IsTriangulated
T : Triangle D
hT : T ∈ distinguishedTriangles
this✝¹ : G.Additive
Z : C
f : G.obj T.obj₂ ⟶ Z
g : Z ⟶ (shiftFunctor C 1).obj (G.obj T.obj₁)
mem : Triangle.mk (G.map T.mor₁) f g ∈ distinguishedTriangles
h : F.obj Z ⟶ T.obj₃
h₁ : F.map f ≫ h = adj.counit.app T.obj₂ ≫ T.mor₂
h₂ :
  F.map g ≫ (Functor.commShiftIso F 1).hom.app (G.obj T.obj₁) ≫ (shiftFunctor D 1).map (adj.counit.app T.obj₁) =
    h ≫ T.mor₃
h₁' : f ≫ adj.unit.app Z ≫ G.map h = G.map T.mor₂
h₂' : g ≫ (Functor.commShiftIso G 1).inv.app T.obj₁ = (adj.homEquiv Z T.obj₃) h ≫ G.map T.mor₃
this✝ : Mono ((adj.homEquiv Z T.obj₃) h)
this : IsIso ((adj.homEquiv Z T.obj₃) h)
⊢ g =
    adj.unit.app Z ≫
      G.map
          (F.map g ≫
            (Functor.commShiftIso F 1).hom.app (G.obj T.obj₁) ≫ (shiftFunctor D 1).map (adj.counit.app T.obj₁)) ≫
        (Functor.commShiftIso G 1).hom.app T.obj₁

Tactic:
Functor.map_comp,

Goal state after:
case refine_3
C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : HasZeroObject C
inst✝¹² : HasZeroObject D
inst✝¹¹ : Preadditive C
inst✝¹⁰ : Preadditive D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : HasShift D ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : Pretriangulated D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝³ : F.CommShift ℤ
inst✝² : G.CommShift ℤ
inst✝¹ : adj.CommShift ℤ
inst✝ : F.IsTriangulated
T : Triangle D
hT : T ∈ distinguishedTriangles
this✝¹ : G.Additive
Z : C
f : G.obj T.obj₂ ⟶ Z
g : Z ⟶ (shiftFunctor C 1).obj (G.obj T.obj₁)
mem : Triangle.mk (G.map T.mor₁) f g ∈ distinguishedTriangles
h : F.obj Z ⟶ T.obj₃
h₁ : F.map f ≫ h = adj.counit.app T.obj₂ ≫ T.mor₂
h₂ :
  F.map g ≫ (Functor.commShiftIso F 1).hom.app (G.obj T.obj₁) ≫ (shiftFunctor D 1).map (adj.counit.app T.obj₁) =
    h ≫ T.mor₃
h₁' : f ≫ adj.unit.app Z ≫ G.map h = G.map T.mor₂
h₂' : g ≫ (Functor.commShiftIso G 1).inv.app T.obj₁ = (adj.homEquiv Z T.obj₃) h ≫ G.map T.mor₃
this✝ : Mono ((adj.homEquiv Z T.obj₃) h)
this : IsIso ((adj.homEquiv Z T.obj₃) h)
⊢ g =
    adj.unit.app Z ≫
      (G.map (F.map g) ≫
          G.map ((Functor.commShiftIso F 1).hom.app (G.obj T.obj₁) ≫ (shiftFunctor D 1).map (adj.counit.app T.obj₁))) ≫
        (Functor.commShiftIso G 1).hom.app T.obj₁