mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Bousfield

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : Abelian C
inst✝⁴ : Abelian D
G : D ⥤ C
inst✝³ : PreservesFiniteLimits G
inst✝² : PreservesFiniteColimits G
F : C ⥤ D
adj : G ⊣ F
inst✝¹ : F.Full
inst✝ : F.Faithful
⊢ G.kernel.isoModSerre = ObjectProperty.isLocal fun x ↦ x ∈ Set.range F.obj

Tactic:
by
  rw [ObjectProperty.isLocal_eq_inverseImage_isomorphisms adj,
    isoModSerre_kernel_eq_inverseImage_isomorphisms]

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.RightDerived

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{v_1, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
n : ℕ
X Y : C
f : X ⟶ Y
P : InjectiveResolution X
Q : InjectiveResolution Y
g : P.cocomplex ⟶ Q.cocomplex
w : P.ι ≫ g = (CochainComplex.single₀ C).map f ≫ Q.ι
⊢ (P.isoRightDerivedObj F n).hom ≫
      (F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomologicalComplex.homologyFunctor D (ComplexShape.up ℕ) n).map g =
    ((P.isoRightDerivedObj F n).hom ≫
        (F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomologicalComplex.homologyFunctor D (ComplexShape.up ℕ) n).map
            g ≫
          (Q.isoRightDerivedObj F n).inv) ≫
      (Q.isoRightDerivedObj F n).hom

pp: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{v_1, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
n : ℕ
X Y : C
f : X ⟶ Y
P : InjectiveResolution X
Q : InjectiveResolution Y
g : P.cocomplex ⟶ Q.cocomplex
w : P.ι ≫ g = (CochainComplex.single₀ C).map f ≫ Q.ι
⊢ P.ι.f 0 ≫ g.f 0 = f ≫ Q.ι.f 0

Tactic:
assoc,

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{v_1, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
n : ℕ
X Y : C
f : X ⟶ Y
P : InjectiveResolution X
Q : InjectiveResolution Y
g : P.cocomplex ⟶ Q.cocomplex
w : P.ι ≫ g = (CochainComplex.single₀ C).map f ≫ Q.ι
⊢ (P.isoRightDerivedObj F n).hom ≫
      (F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomologicalComplex.homologyFunctor D (ComplexShape.up ℕ) n).map g =
    (P.isoRightDerivedObj F n).hom ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomologicalComplex.homologyFunctor D (ComplexShape.up ℕ) n).map
            g ≫
          (Q.isoRightDerivedObj F n).inv) ≫
        (Q.isoRightDerivedObj F n).hom

C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{v_1, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
n : ℕ
X Y : C
f : X ⟶ Y
P : InjectiveResolution X
Q : InjectiveResolution Y
g : P.cocomplex ⟶ Q.cocomplex
w : P.ι ≫ g = (CochainComplex.single₀ C).map f ≫ Q.ι
⊢ P.ι.f 0 ≫ g.f 0 = f ≫ Q.ι.f 0
Module: Mathlib.CategoryTheory.Abelian.RightDerived

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{v_1, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
n : ℕ
X Y : C
f : X ⟶ Y
P : InjectiveResolution X
Q : InjectiveResolution Y
g : P.cocomplex ⟶ Q.cocomplex
w : P.ι ≫ g = (CochainComplex.single₀ C).map f ≫ Q.ι
⊢ (P.isoRightDerivedObj F n).hom ≫
      (F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomologicalComplex.homologyFunctor D (ComplexShape.up ℕ) n).map g =
    (P.isoRightDerivedObj F n).hom ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomologicalComplex.homologyFunctor D (ComplexShape.up ℕ) n).map
            g ≫
          (Q.isoRightDerivedObj F n).inv) ≫
        (Q.isoRightDerivedObj F n).hom

pp: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{v_1, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
n : ℕ
X Y : C
f : X ⟶ Y
P : InjectiveResolution X
Q : InjectiveResolution Y
g : P.cocomplex ⟶ Q.cocomplex
w : P.ι ≫ g = (CochainComplex.single₀ C).map f ≫ Q.ι
⊢ P.ι.f 0 ≫ g.f 0 = f ≫ Q.ι.f 0

Tactic:
assoc,

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{v_1, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
n : ℕ
X Y : C
f : X ⟶ Y
P : InjectiveResolution X
Q : InjectiveResolution Y
g : P.cocomplex ⟶ Q.cocomplex
w : P.ι ≫ g = (CochainComplex.single₀ C).map f ≫ Q.ι
⊢ (P.isoRightDerivedObj F n).hom ≫
      (F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomologicalComplex.homologyFunctor D (ComplexShape.up ℕ) n).map g =
    (P.isoRightDerivedObj F n).hom ≫
      (F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomologicalComplex.homologyFunctor D (ComplexShape.up ℕ) n).map g ≫
        (Q.isoRightDerivedObj F n).inv ≫ (Q.isoRightDerivedObj F n).hom

C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{v_1, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
n : ℕ
X Y : C
f : X ⟶ Y
P : InjectiveResolution X
Q : InjectiveResolution Y
g : P.cocomplex ⟶ Q.cocomplex
w : P.ι ≫ g = (CochainComplex.single₀ C).map f ≫ Q.ι
⊢ P.ι.f 0 ≫ g.f 0 = f ≫ Q.ι.f 0
Module: Mathlib.CategoryTheory.Abelian.RightDerived

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{v_1, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasInjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : InjectiveResolution X
⊢ (NatTrans.rightDerivedToHomotopyCategory α).app X ≫ (P.isoRightDerivedToHomotopyCategoryObj G).hom =
    ((P.isoRightDerivedToHomotopyCategoryObj F).hom ≫
        (HomotopyCategory.quotient D (ComplexShape.up ℕ)).map
            ((NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app P.cocomplex) ≫
          (P.isoRightDerivedToHomotopyCategoryObj G).inv) ≫
      (P.isoRightDerivedToHomotopyCategoryObj G).hom

Tactic:
assoc,

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{v_1, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasInjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : InjectiveResolution X
⊢ (NatTrans.rightDerivedToHomotopyCategory α).app X ≫ (P.isoRightDerivedToHomotopyCategoryObj G).hom =
    (P.isoRightDerivedToHomotopyCategoryObj F).hom ≫
      ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).map
            ((NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app P.cocomplex) ≫
          (P.isoRightDerivedToHomotopyCategoryObj G).inv) ≫
        (P.isoRightDerivedToHomotopyCategoryObj G).hom
Module: Mathlib.CategoryTheory.Abelian.RightDerived

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{v_1, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasInjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : InjectiveResolution X
⊢ (NatTrans.rightDerivedToHomotopyCategory α).app X ≫ (P.isoRightDerivedToHomotopyCategoryObj G).hom =
    (P.isoRightDerivedToHomotopyCategoryObj F).hom ≫
      ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).map
            ((NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app P.cocomplex) ≫
          (P.isoRightDerivedToHomotopyCategoryObj G).inv) ≫
        (P.isoRightDerivedToHomotopyCategoryObj G).hom

Tactic:
assoc,

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{v_1, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasInjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : InjectiveResolution X
⊢ (NatTrans.rightDerivedToHomotopyCategory α).app X ≫ (P.isoRightDerivedToHomotopyCategoryObj G).hom =
    (P.isoRightDerivedToHomotopyCategoryObj F).hom ≫
      (HomotopyCategory.quotient D (ComplexShape.up ℕ)).map
          ((NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app P.cocomplex) ≫
        (P.isoRightDerivedToHomotopyCategoryObj G).inv ≫ (P.isoRightDerivedToHomotopyCategoryObj G).hom
Module: Mathlib.CategoryTheory.Abelian.RightDerived

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{v_1, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
X Y : C
f : X ⟶ Y
⊢ F.map f ≫
      (injectiveResolution Y).toRightDerivedZero' F ≫
        ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution Y).cocomplex).homologyπ 0 ≫
          (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution Y).cocomplex) =
    ((injectiveResolution X).toRightDerivedZero' F ≫
        ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫
          (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex)) ≫
      (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) 0).map (F.rightDerivedToHomotopyCategory.map f)

Tactic:
assoc,

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{v_1, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
X Y : C
f : X ⟶ Y
⊢ F.map f ≫
      (injectiveResolution Y).toRightDerivedZero' F ≫
        ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution Y).cocomplex).homologyπ 0 ≫
          (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution Y).cocomplex) =
    (injectiveResolution X).toRightDerivedZero' F ≫
      (((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫
          (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex)) ≫
        (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) 0).map (F.rightDerivedToHomotopyCategory.map f)
Module: Mathlib.CategoryTheory.Abelian.RightDerived

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{v_1, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
X Y : C
f : X ⟶ Y
⊢ F.map f ≫
      (injectiveResolution Y).toRightDerivedZero' F ≫
        ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution Y).cocomplex).homologyπ 0 ≫
          (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution Y).cocomplex) =
    (injectiveResolution X).toRightDerivedZero' F ≫
      (((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫
          (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex)) ≫
        (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) 0).map (F.rightDerivedToHomotopyCategory.map f)

Tactic:
assoc,

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{v_1, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
F : C ⥤ D
inst✝ : F.Additive
X Y : C
f : X ⟶ Y
⊢ F.map f ≫
      (injectiveResolution Y).toRightDerivedZero' F ≫
        ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution Y).cocomplex).homologyπ 0 ≫
          (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution Y).cocomplex) =
    (injectiveResolution X).toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫
        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex) ≫
          (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) 0).map (F.rightDerivedToHomotopyCategory.map f)
Module: Mathlib.CategoryTheory.ObjectProperty.Kernels

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
P : ObjectProperty C
inst✝ : P.IsClosedUnderSubobjects
i✝ X₁✝ X₂✝ : C
f✝ : X₁✝ ⟶ X₂✝
k : KernelFork f✝
hk : IsLimit k
hf : MorphismProperty.ofObjectProperty P P f✝
this : Mono (Fork.ι k) := Fork.IsLimit.mono hk
⊢ P k.pt

Tactic:
exact P.prop_of_mono k.ι hf.1

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCategory

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Groupoid C
inst✝² : Category.{v_1, u_2} D
inst✝¹ : MonoidalCategory D
inst✝ : RightRigidCategory D
F : C ⥤ D
X Y : C
f : X ⟶ Y
⊢ η_ (F.obj Y) (F.obj Y)ᘁ = η_ (F.obj X) (F.obj X)ᘁ ≫ (𝟙 (F.obj X) ⊗ₘ inv (F.map f)ᘁ) ≫ (F.map f ⊗ₘ 𝟙 (F.obj Y)ᘁ)

Tactic:
id_tensorHom,

Goal state after:
C : Type u_1
D : Type u_2
inst✝³ : Groupoid C
inst✝² : Category.{v_1, u_2} D
inst✝¹ : MonoidalCategory D
inst✝ : RightRigidCategory D
F : C ⥤ D
X Y : C
f : X ⟶ Y
⊢ η_ (F.obj Y) (F.obj Y)ᘁ = η_ (F.obj X) (F.obj X)ᘁ ≫ F.obj X ◁ inv (F.map f)ᘁ ≫ (F.map f ⊗ₘ 𝟙 (F.obj Y)ᘁ)
Module: Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCategory

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Groupoid C
inst✝² : Category.{v_1, u_2} D
inst✝¹ : MonoidalCategory D
inst✝ : RightRigidCategory D
F : C ⥤ D
X Y : C
f : X ⟶ Y
⊢ (𝟙 (F.obj X)ᘁ ⊗ₘ F.map f) ≫ (inv (F.map f)ᘁ ⊗ₘ 𝟙 (F.obj Y)) ≫ ε_ (F.obj Y) (F.obj Y)ᘁ = ε_ (F.obj X) (F.obj X)ᘁ

Tactic:
id_tensorHom,

Goal state after:
C : Type u_1
D : Type u_2
inst✝³ : Groupoid C
inst✝² : Category.{v_1, u_2} D
inst✝¹ : MonoidalCategory D
inst✝ : RightRigidCategory D
F : C ⥤ D
X Y : C
f : X ⟶ Y
⊢ (F.obj X)ᘁ ◁ F.map f ≫ (inv (F.map f)ᘁ ⊗ₘ 𝟙 (F.obj Y)) ≫ ε_ (F.obj Y) (F.obj Y)ᘁ = ε_ (F.obj X) (F.obj X)ᘁ
Module: Mathlib.CategoryTheory.Adjunction.Lifting.Right

Goal state before:
pp: case refine_1
A : Type u₁
B : Type u₂
C : Type u₃
inst✝² : Category.{v₁, u₁} A
inst✝¹ : Category.{v₂, u₂} B
inst✝ : Category.{v₃, u₃} C
U : A ⥤ B
F : B ⥤ A
L : C ⥤ B
U' : A ⥤ C
adj₁ : F ⊣ U
adj₂ : L ⋙ F ⊣ U'
h : (X : B) → RegularMono (adj₁.unit.app X)
X : B
s : Fork (U.map (F.map (adj₁.unit.app X))) (adj₁.unit.app ((F ⋙ U).obj X))
this : ∀ (Y : B), Mono (adj₁.unit.app Y)
⊢ (s.ι ≫ (h X).left) ≫ adj₁.unit.app (h X).Z = (s.ι ≫ (h X).right) ≫ adj₁.unit.app (h X).Z

Tactic:
assoc,

Goal state after:
case refine_1
A : Type u₁
B : Type u₂
C : Type u₃
inst✝² : Category.{v₁, u₁} A
inst✝¹ : Category.{v₂, u₂} B
inst✝ : Category.{v₃, u₃} C
U : A ⥤ B
F : B ⥤ A
L : C ⥤ B
U' : A ⥤ C
adj₁ : F ⊣ U
adj₂ : L ⋙ F ⊣ U'
h : (X : B) → RegularMono (adj₁.unit.app X)
X : B
s : Fork (U.map (F.map (adj₁.unit.app X))) (adj₁.unit.app ((F ⋙ U).obj X))
this : ∀ (Y : B), Mono (adj₁.unit.app Y)
⊢ s.ι ≫ (h X).left ≫ adj₁.unit.app (h X).Z = (s.ι ≫ (h X).right) ≫ adj₁.unit.app (h X).Z
Module: Mathlib.CategoryTheory.Adjunction.Lifting.Right

Goal state before:
pp: case refine_1
A : Type u₁
B : Type u₂
C : Type u₃
inst✝² : Category.{v₁, u₁} A
inst✝¹ : Category.{v₂, u₂} B
inst✝ : Category.{v₃, u₃} C
U : A ⥤ B
F : B ⥤ A
L : C ⥤ B
U' : A ⥤ C
adj₁ : F ⊣ U
adj₂ : L ⋙ F ⊣ U'
h : (X : B) → RegularMono (adj₁.unit.app X)
X : B
s : Fork (U.map (F.map (adj₁.unit.app X))) (adj₁.unit.app ((F ⋙ U).obj X))
this✝ : ∀ (Y : B), Mono (adj₁.unit.app Y)
this : s.ι ≫ U.map (F.map (adj₁.unit.app X)) = s.ι ≫ adj₁.unit.app (U.obj (F.obj X))
⊢ (s.ι ≫ U.map (F.map (adj₁.unit.app X))) ≫ U.map (F.map (h X).left) = (s.ι ≫ (h X).right) ≫ adj₁.unit.app (h X).Z

Tactic:
assoc,

Goal state after:
case refine_1
A : Type u₁
B : Type u₂
C : Type u₃
inst✝² : Category.{v₁, u₁} A
inst✝¹ : Category.{v₂, u₂} B
inst✝ : Category.{v₃, u₃} C
U : A ⥤ B
F : B ⥤ A
L : C ⥤ B
U' : A ⥤ C
adj₁ : F ⊣ U
adj₂ : L ⋙ F ⊣ U'
h : (X : B) → RegularMono (adj₁.unit.app X)
X : B
s : Fork (U.map (F.map (adj₁.unit.app X))) (adj₁.unit.app ((F ⋙ U).obj X))
this✝ : ∀ (Y : B), Mono (adj₁.unit.app Y)
this : s.ι ≫ U.map (F.map (adj₁.unit.app X)) = s.ι ≫ adj₁.unit.app (U.obj (F.obj X))
⊢ s.ι ≫ U.map (F.map (adj₁.unit.app X)) ≫ U.map (F.map (h X).left) = (s.ι ≫ (h X).right) ≫ adj₁.unit.app (h X).Z
Module: Mathlib.CategoryTheory.Adjunction.Lifting.Right

Goal state before:
pp: case refine_1
A : Type u₁
B : Type u₂
C : Type u₃
inst✝² : Category.{v₁, u₁} A
inst✝¹ : Category.{v₂, u₂} B
inst✝ : Category.{v₃, u₃} C
U : A ⥤ B
F : B ⥤ A
L : C ⥤ B
U' : A ⥤ C
adj₁ : F ⊣ U
adj₂ : L ⋙ F ⊣ U'
h : (X : B) → RegularMono (adj₁.unit.app X)
X : B
s : Fork (U.map (F.map (adj₁.unit.app X))) (adj₁.unit.app ((F ⋙ U).obj X))
this✝ : ∀ (Y : B), Mono (adj₁.unit.app Y)
this : s.ι ≫ U.map (F.map (adj₁.unit.app X)) = s.ι ≫ adj₁.unit.app (U.obj (F.obj X))
⊢ s.ι ≫ adj₁.unit.app ((F ⋙ U).obj X) ≫ U.map (F.map (h X).right) = (s.ι ≫ (h X).right) ≫ adj₁.unit.app (h X).Z

Tactic:
assoc,

Goal state after:
case refine_1
A : Type u₁
B : Type u₂
C : Type u₃
inst✝² : Category.{v₁, u₁} A
inst✝¹ : Category.{v₂, u₂} B
inst✝ : Category.{v₃, u₃} C
U : A ⥤ B
F : B ⥤ A
L : C ⥤ B
U' : A ⥤ C
adj₁ : F ⊣ U
adj₂ : L ⋙ F ⊣ U'
h : (X : B) → RegularMono (adj₁.unit.app X)
X : B
s : Fork (U.map (F.map (adj₁.unit.app X))) (adj₁.unit.app ((F ⋙ U).obj X))
this✝ : ∀ (Y : B), Mono (adj₁.unit.app Y)
this : s.ι ≫ U.map (F.map (adj₁.unit.app X)) = s.ι ≫ adj₁.unit.app (U.obj (F.obj X))
⊢ s.ι ≫ adj₁.unit.app ((F ⋙ U).obj X) ≫ U.map (F.map (h X).right) = s.ι ≫ (h X).right ≫ adj₁.unit.app (h X).Z
Module: Mathlib.CategoryTheory.Adjunction.Lifting.Left

Goal state before:
pp: A : Type u₁
B : Type u₂
C : Type u₃
inst✝³ : Category.{v₁, u₁} A
inst✝² : Category.{v₂, u₂} B
inst✝¹ : Category.{v₃, u₃} C
U : B ⥤ C
F : C ⥤ B
R : A ⥤ B
F' : C ⥤ A
adj₁ : F ⊣ U
adj₂ : F' ⊣ R ⋙ U
inst✝ : HasReflexiveCoequalizers A
h : (X : B) → RegularEpi (adj₁.counit.app X)
Y : A
X : B
f : F'.obj (U.obj X) ⟶ Y
⊢ (adj₂.homEquiv (U.obj ((U ⋙ F).obj X)) Y)
        ((F'.map (U.map (F.map (adj₂.unit.app (U.1 X)) ≫ adj₁.counit.app (R.obj (F'.obj (U.1 X))))) ≫
            adj₂.counit.app (F'.obj (U.1 X))) ≫
          f) =
      U.map (adj₁.counit.app X) ≫ (adj₂.homEquiv (U.obj ((𝟭 B).obj X)) Y) f ↔
    U.map (F.map ((adj₂.homEquiv (U.obj X) Y) f) ≫ adj₁.counit.app (R.obj Y)) =
      U.map (adj₁.counit.app X) ≫ (adj₂.homEquiv (U.obj X) Y) f

Tactic:
assoc,

Goal state after:
A : Type u₁
B : Type u₂
C : Type u₃
inst✝³ : Category.{v₁, u₁} A
inst✝² : Category.{v₂, u₂} B
inst✝¹ : Category.{v₃, u₃} C
U : B ⥤ C
F : C ⥤ B
R : A ⥤ B
F' : C ⥤ A
adj₁ : F ⊣ U
adj₂ : F' ⊣ R ⋙ U
inst✝ : HasReflexiveCoequalizers A
h : (X : B) → RegularEpi (adj₁.counit.app X)
Y : A
X : B
f : F'.obj (U.obj X) ⟶ Y
⊢ (adj₂.homEquiv (U.obj ((U ⋙ F).obj X)) Y)
        (F'.map (U.map (F.map (adj₂.unit.app (U.1 X)) ≫ adj₁.counit.app (R.obj (F'.obj (U.1 X))))) ≫
          adj₂.counit.app (F'.obj (U.1 X)) ≫ f) =
      U.map (adj₁.counit.app X) ≫ (adj₂.homEquiv (U.obj ((𝟭 B).obj X)) Y) f ↔
    U.map (F.map ((adj₂.homEquiv (U.obj X) Y) f) ≫ adj₁.counit.app (R.obj Y)) =
      U.map (adj₁.counit.app X) ≫ (adj₂.homEquiv (U.obj X) Y) f
Module: Mathlib.CategoryTheory.Adjunction.Lifting.Left

Goal state before:
pp: A : Type u₁
B : Type u₂
C : Type u₃
inst✝³ : Category.{v₁, u₁} A
inst✝² : Category.{v₂, u₂} B
inst✝¹ : Category.{v₃, u₃} C
U : B ⥤ C
F : C ⥤ B
R : A ⥤ B
F' : C ⥤ A
adj₁ : F ⊣ U
adj₂ : F' ⊣ R ⋙ U
inst✝ : HasReflexiveCoequalizers A
h : (X : B) → RegularEpi (adj₁.counit.app X)
Y : A
X : B
g : U.obj X ⟶ U.obj (R.obj Y)
⊢ F.map (U.map (F.map g ≫ adj₁.counit.app (R.obj Y))) ≫ adj₁.counit.app ((𝟭 B).obj (R.obj Y)) =
      (F.map (U.map (adj₁.counit.app X)) ≫ F.map g) ≫ adj₁.counit.app ((𝟭 B).obj (R.obj Y)) ↔
    F.map (U.map (adj₁.counit.app X)) ≫ F.map g ≫ adj₁.counit.app (R.obj Y) =
      adj₁.counit.app ((U ⋙ F).obj X) ≫ F.map g ≫ adj₁.counit.app (R.obj Y)

Tactic:
assoc,

Goal state after:
A : Type u₁
B : Type u₂
C : Type u₃
inst✝³ : Category.{v₁, u₁} A
inst✝² : Category.{v₂, u₂} B
inst✝¹ : Category.{v₃, u₃} C
U : B ⥤ C
F : C ⥤ B
R : A ⥤ B
F' : C ⥤ A
adj₁ : F ⊣ U
adj₂ : F' ⊣ R ⋙ U
inst✝ : HasReflexiveCoequalizers A
h : (X : B) → RegularEpi (adj₁.counit.app X)
Y : A
X : B
g : U.obj X ⟶ U.obj (R.obj Y)
⊢ F.map (U.map (F.map g ≫ adj₁.counit.app (R.obj Y))) ≫ adj₁.counit.app ((𝟭 B).obj (R.obj Y)) =
      F.map (U.map (adj₁.counit.app X)) ≫ F.map g ≫ adj₁.counit.app ((𝟭 B).obj (R.obj Y)) ↔
    F.map (U.map (adj₁.counit.app X)) ≫ F.map g ≫ adj₁.counit.app (R.obj Y) =
      adj₁.counit.app ((U ⋙ F).obj X) ≫ F.map g ≫ adj₁.counit.app (R.obj Y)
Module: Mathlib.CategoryTheory.Adjunction.Lifting.Left

Goal state before:
pp: A : Type u₁
B : Type u₂
C : Type u₃
inst✝³ : Category.{v₁, u₁} A
inst✝² : Category.{v₂, u₂} B
inst✝¹ : Category.{v₃, u₃} C
U : B ⥤ C
F : C ⥤ B
R : A ⥤ B
F' : C ⥤ A
adj₁ : F ⊣ U
adj₂ : F' ⊣ R ⋙ U
inst✝ : HasReflexiveCoequalizers A
h : (X : B) → RegularEpi (adj₁.counit.app X)
Y : A
X : B
g : U.obj X ⟶ U.obj (R.obj Y)
⊢ (F.map (U.map (F.map g)) ≫ F.map (U.map (adj₁.counit.app (R.obj Y)))) ≫ adj₁.counit.app ((𝟭 B).obj (R.obj Y)) =
      F.map (U.map (adj₁.counit.app X)) ≫ F.map g ≫ adj₁.counit.app ((𝟭 B).obj (R.obj Y)) ↔
    F.map (U.map (adj₁.counit.app X)) ≫ F.map g ≫ adj₁.counit.app (R.obj Y) =
      adj₁.counit.app ((U ⋙ F).obj X) ≫ F.map g ≫ adj₁.counit.app (R.obj Y)

Tactic:
assoc,

Goal state after:
A : Type u₁
B : Type u₂
C : Type u₃
inst✝³ : Category.{v₁, u₁} A
inst✝² : Category.{v₂, u₂} B
inst✝¹ : Category.{v₃, u₃} C
U : B ⥤ C
F : C ⥤ B
R : A ⥤ B
F' : C ⥤ A
adj₁ : F ⊣ U
adj₂ : F' ⊣ R ⋙ U
inst✝ : HasReflexiveCoequalizers A
h : (X : B) → RegularEpi (adj₁.counit.app X)
Y : A
X : B
g : U.obj X ⟶ U.obj (R.obj Y)
⊢ F.map (U.map (F.map g)) ≫ F.map (U.map (adj₁.counit.app (R.obj Y))) ≫ adj₁.counit.app ((𝟭 B).obj (R.obj Y)) =
      F.map (U.map (adj₁.counit.app X)) ≫ F.map g ≫ adj₁.counit.app ((𝟭 B).obj (R.obj Y)) ↔
    F.map (U.map (adj₁.counit.app X)) ≫ F.map g ≫ adj₁.counit.app (R.obj Y) =
      adj₁.counit.app ((U ⋙ F).obj X) ≫ F.map g ≫ adj₁.counit.app (R.obj Y)
Module: Mathlib.CategoryTheory.Adjunction.Quadruple

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
L : C ⥤ D
F : D ⥤ C
G : C ⥤ D
R : D ⥤ C
q : Quadruple L F G R
inst✝¹ : F.Full
inst✝ : F.Faithful
X : D
Y : C
⊢ Function.Injective
      (⇑((q.adj₂.homEquiv (G.obj Y) (F.obj X)).trans (q.adj₃.homEquiv Y (G.obj (F.obj X)))) ∘ fun f ↦
        q.adj₂.counit.app Y ≫ f) ↔
    Function.Injective fun g ↦ g ≫ q.adj₃.unit.app (F.obj X)

Tactic:
simp [Function.comp_def, ← q.adj₂.homEquiv_symm_id, ← q.adj₂.homEquiv_naturality_right_symm,
    ← q.adj₃.homEquiv_id, ← q.adj₃.homEquiv_naturality_left]

Goal state after:
Module: Mathlib.CategoryTheory.Adjunction.Lifting.Left

Goal state before:
pp: A : Type u₁
B : Type u₂
C : Type u₃
inst✝⁵ : Category.{v₁, u₁} A
inst✝⁴ : Category.{v₂, u₂} B
inst✝³ : Category.{v₃, u₃} C
U : B ⥤ C
inst✝² : MonadicRightAdjoint U
R : A ⥤ B
inst✝¹ : HasReflexiveCoequalizers A
inst✝ : (R ⋙ U).IsRightAdjoint
R' : A ⥤ (monadicAdjunction U).toMonad.Algebra := R ⋙ Monad.comparison (monadicAdjunction U)
this : (R' ⋙ (monadicAdjunction U).toMonad.forget).IsRightAdjoint :=
  Adjunction.isRightAdjoint
    ((Adjunction.ofIsRightAdjoint (R ⋙ U)).ofNatIsoRight
      (R.isoWhiskerLeft (Monad.comparisonForget (monadicAdjunction U)).symm))
X : (monadicAdjunction U).toMonad.Algebra
⊢ RegularEpi ((monadicAdjunction U).toMonad.adj.counit.app X)

Tactic:
simp only [Monad.adj_counit]

Goal state after:
A : Type u₁
B : Type u₂
C : Type u₃
inst✝⁵ : Category.{v₁, u₁} A
inst✝⁴ : Category.{v₂, u₂} B
inst✝³ : Category.{v₃, u₃} C
U : B ⥤ C
inst✝² : MonadicRightAdjoint U
R : A ⥤ B
inst✝¹ : HasReflexiveCoequalizers A
inst✝ : (R ⋙ U).IsRightAdjoint
R' : A ⥤ (monadicAdjunction U).toMonad.Algebra := R ⋙ Monad.comparison (monadicAdjunction U)
this : (R' ⋙ (monadicAdjunction U).toMonad.forget).IsRightAdjoint :=
  Adjunction.isRightAdjoint
    ((Adjunction.ofIsRightAdjoint (R ⋙ U)).ofNatIsoRight
      (R.isoWhiskerLeft (Monad.comparisonForget (monadicAdjunction U)).symm))
X : (monadicAdjunction U).toMonad.Algebra
⊢ RegularEpi { f := X.a, h := ⋯ }