Module: Mathlib.CategoryTheory.Shift.Adjunction

Goal state before:
pp: case a.a.a.a.a.a.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
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a b : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
f₁ : shiftFunctor C b ⋙ F ≅ F ⋙ shiftFunctor D b
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
f₂ : shiftFunctor D b ⋙ G ≅ G ⋙ shiftFunctor C b
h : CompatibilityUnit adj e₁ e₂
h' : CompatibilityUnit adj f₁ f₂
X : C
this :
  (shiftFunctor C b).map (adj.unit.app ((shiftFunctor C a).obj X)) ≫ f₂.inv.app (F.obj ((shiftFunctor C a).obj X)) =
    adj.unit.app ((shiftFunctor C b).obj ((shiftFunctor C a).obj X)) ≫ G.map (f₁.hom.app ((shiftFunctor C a).obj X))
| G.map ((shiftFunctorAdd D a b).inv.app (F.obj X)) ≫ G.map ((shiftFunctorAdd D a b).hom.app (F.obj X))

pp: case a.a.a.a.a.a.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
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a b : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
f₁ : shiftFunctor C b ⋙ F ≅ F ⋙ shiftFunctor D b
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
f₂ : shiftFunctor D b ⋙ G ≅ G ⋙ shiftFunctor C b
h : CompatibilityUnit adj e₁ e₂
h' : CompatibilityUnit adj f₁ f₂
X : C
this :
  (shiftFunctor C b).map (adj.unit.app ((shiftFunctor C a).obj X)) ≫ f₂.inv.app (F.obj ((shiftFunctor C a).obj X)) =
    adj.unit.app ((shiftFunctor C b).obj ((shiftFunctor C a).obj X)) ≫ G.map (f₁.hom.app ((shiftFunctor C a).obj X))
| f₂.hom.app ((shiftFunctor D a).obj (F.obj X))

pp: case a.a.a.a.a.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
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a b : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
f₁ : shiftFunctor C b ⋙ F ≅ F ⋙ shiftFunctor D b
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
f₂ : shiftFunctor D b ⋙ G ≅ G ⋙ shiftFunctor C b
h : CompatibilityUnit adj e₁ e₂
h' : CompatibilityUnit adj f₁ f₂
X : C
this :
  (shiftFunctor C b).map (adj.unit.app ((shiftFunctor C a).obj X)) ≫ f₂.inv.app (F.obj ((shiftFunctor C a).obj X)) =
    adj.unit.app ((shiftFunctor C b).obj ((shiftFunctor C a).obj X)) ≫ G.map (f₁.hom.app ((shiftFunctor C a).obj X))
| (shiftFunctor C b).map (e₂.hom.app (F.obj X))

pp: case a.a.a.a.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
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a b : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
f₁ : shiftFunctor C b ⋙ F ≅ F ⋙ shiftFunctor D b
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
f₂ : shiftFunctor D b ⋙ G ≅ G ⋙ shiftFunctor C b
h : CompatibilityUnit adj e₁ e₂
h' : CompatibilityUnit adj f₁ f₂
X : C
this :
  (shiftFunctor C b).map (adj.unit.app ((shiftFunctor C a).obj X)) ≫ f₂.inv.app (F.obj ((shiftFunctor C a).obj X)) =
    adj.unit.app ((shiftFunctor C b).obj ((shiftFunctor C a).obj X)) ≫ G.map (f₁.hom.app ((shiftFunctor C a).obj X))
| (shiftFunctorAdd C a b).inv.app (G.obj (F.obj X))

pp: case 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
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a b : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
f₁ : shiftFunctor C b ⋙ F ≅ F ⋙ shiftFunctor D b
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
f₂ : shiftFunctor D b ⋙ G ≅ G ⋙ shiftFunctor C b
h : CompatibilityUnit adj e₁ e₂
h' : CompatibilityUnit adj f₁ f₂
X : C
this :
  (shiftFunctor C b).map (adj.unit.app ((shiftFunctor C a).obj X)) ≫ f₂.inv.app (F.obj ((shiftFunctor C a).obj X)) =
    adj.unit.app ((shiftFunctor C b).obj ((shiftFunctor C a).obj X)) ≫ G.map (f₁.hom.app ((shiftFunctor C a).obj X))
| (shiftFunctorAdd C a b).hom.app X

pp: case a.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
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a b : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
f₁ : shiftFunctor C b ⋙ F ≅ F ⋙ shiftFunctor D b
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
f₂ : shiftFunctor D b ⋙ G ≅ G ⋙ shiftFunctor C b
h : CompatibilityUnit adj e₁ e₂
h' : CompatibilityUnit adj f₁ f₂
X : C
this :
  (shiftFunctor C b).map (adj.unit.app ((shiftFunctor C a).obj X)) ≫ f₂.inv.app (F.obj ((shiftFunctor C a).obj X)) =
    adj.unit.app ((shiftFunctor C b).obj ((shiftFunctor C a).obj X)) ≫ G.map (f₁.hom.app ((shiftFunctor C a).obj X))
| adj.unit.app ((shiftFunctor C b).obj ((shiftFunctor C a).obj X))

pp: case a.a.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
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a b : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
f₁ : shiftFunctor C b ⋙ F ≅ F ⋙ shiftFunctor D b
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
f₂ : shiftFunctor D b ⋙ G ≅ G ⋙ shiftFunctor C b
h : CompatibilityUnit adj e₁ e₂
h' : CompatibilityUnit adj f₁ f₂
X : C
this :
  (shiftFunctor C b).map (adj.unit.app ((shiftFunctor C a).obj X)) ≫ f₂.inv.app (F.obj ((shiftFunctor C a).obj X)) =
    adj.unit.app ((shiftFunctor C b).obj ((shiftFunctor C a).obj X)) ≫ G.map (f₁.hom.app ((shiftFunctor C a).obj X))
| G.map (f₁.hom.app ((shiftFunctor C a).obj X))

pp: case a.a.a.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
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a b : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
f₁ : shiftFunctor C b ⋙ F ≅ F ⋙ shiftFunctor D b
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
f₂ : shiftFunctor D b ⋙ G ≅ G ⋙ shiftFunctor C b
h : CompatibilityUnit adj e₁ e₂
h' : CompatibilityUnit adj f₁ f₂
X : C
this :
  (shiftFunctor C b).map (adj.unit.app ((shiftFunctor C a).obj X)) ≫ f₂.inv.app (F.obj ((shiftFunctor C a).obj X)) =
    adj.unit.app ((shiftFunctor C b).obj ((shiftFunctor C a).obj X)) ≫ G.map (f₁.hom.app ((shiftFunctor C a).obj X))
| G.map ((shiftFunctor D b).map (e₁.hom.app X))

Tactic:
rw [← G.map_comp, Iso.inv_hom_id_app]

Goal state after:
case a.a.a.a.a.a.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
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a b : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
f₁ : shiftFunctor C b ⋙ F ≅ F ⋙ shiftFunctor D b
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
f₂ : shiftFunctor D b ⋙ G ≅ G ⋙ shiftFunctor C b
h : CompatibilityUnit adj e₁ e₂
h' : CompatibilityUnit adj f₁ f₂
X : C
this :
  (shiftFunctor C b).map (adj.unit.app ((shiftFunctor C a).obj X)) ≫ f₂.inv.app (F.obj ((shiftFunctor C a).obj X)) =
    adj.unit.app ((shiftFunctor C b).obj ((shiftFunctor C a).obj X)) ≫ G.map (f₁.hom.app ((shiftFunctor C a).obj X))
| G.map (𝟙 ((shiftFunctor D a ⋙ shiftFunctor D b).obj (F.obj X)))

case a.a.a.a.a.a.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
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a b : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
f₁ : shiftFunctor C b ⋙ F ≅ F ⋙ shiftFunctor D b
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
f₂ : shiftFunctor D b ⋙ G ≅ G ⋙ shiftFunctor C b
h : CompatibilityUnit adj e₁ e₂
h' : CompatibilityUnit adj f₁ f₂
X : C
this :
  (shiftFunctor C b).map (adj.unit.app ((shiftFunctor C a).obj X)) ≫ f₂.inv.app (F.obj ((shiftFunctor C a).obj X)) =
    adj.unit.app ((shiftFunctor C b).obj ((shiftFunctor C a).obj X)) ≫ G.map (f₁.hom.app ((shiftFunctor C a).obj X))
| f₂.hom.app ((shiftFunctor D a).obj (F.obj X))

case a.a.a.a.a.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
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a b : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
f₁ : shiftFunctor C b ⋙ F ≅ F ⋙ shiftFunctor D b
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
f₂ : shiftFunctor D b ⋙ G ≅ G ⋙ shiftFunctor C b
h : CompatibilityUnit adj e₁ e₂
h' : CompatibilityUnit adj f₁ f₂
X : C
this :
  (shiftFunctor C b).map (adj.unit.app ((shiftFunctor C a).obj X)) ≫ f₂.inv.app (F.obj ((shiftFunctor C a).obj X)) =
    adj.unit.app ((shiftFunctor C b).obj ((shiftFunctor C a).obj X)) ≫ G.map (f₁.hom.app ((shiftFunctor C a).obj X))
| (shiftFunctor C b).map (e₂.hom.app (F.obj X))

case a.a.a.a.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
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a b : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
f₁ : shiftFunctor C b ⋙ F ≅ F ⋙ shiftFunctor D b
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
f₂ : shiftFunctor D b ⋙ G ≅ G ⋙ shiftFunctor C b
h : CompatibilityUnit adj e₁ e₂
h' : CompatibilityUnit adj f₁ f₂
X : C
this :
  (shiftFunctor C b).map (adj.unit.app ((shiftFunctor C a).obj X)) ≫ f₂.inv.app (F.obj ((shiftFunctor C a).obj X)) =
    adj.unit.app ((shiftFunctor C b).obj ((shiftFunctor C a).obj X)) ≫ G.map (f₁.hom.app ((shiftFunctor C a).obj X))
| (shiftFunctorAdd C a b).inv.app (G.obj (F.obj X))

case 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
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a b : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
f₁ : shiftFunctor C b ⋙ F ≅ F ⋙ shiftFunctor D b
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
f₂ : shiftFunctor D b ⋙ G ≅ G ⋙ shiftFunctor C b
h : CompatibilityUnit adj e₁ e₂
h' : CompatibilityUnit adj f₁ f₂
X : C
this :
  (shiftFunctor C b).map (adj.unit.app ((shiftFunctor C a).obj X)) ≫ f₂.inv.app (F.obj ((shiftFunctor C a).obj X)) =
    adj.unit.app ((shiftFunctor C b).obj ((shiftFunctor C a).obj X)) ≫ G.map (f₁.hom.app ((shiftFunctor C a).obj X))
| (shiftFunctorAdd C a b).hom.app X

case a.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
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a b : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
f₁ : shiftFunctor C b ⋙ F ≅ F ⋙ shiftFunctor D b
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
f₂ : shiftFunctor D b ⋙ G ≅ G ⋙ shiftFunctor C b
h : CompatibilityUnit adj e₁ e₂
h' : CompatibilityUnit adj f₁ f₂
X : C
this :
  (shiftFunctor C b).map (adj.unit.app ((shiftFunctor C a).obj X)) ≫ f₂.inv.app (F.obj ((shiftFunctor C a).obj X)) =
    adj.unit.app ((shiftFunctor C b).obj ((shiftFunctor C a).obj X)) ≫ G.map (f₁.hom.app ((shiftFunctor C a).obj X))
| adj.unit.app ((shiftFunctor C b).obj ((shiftFunctor C a).obj X))

case a.a.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
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a b : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
f₁ : shiftFunctor C b ⋙ F ≅ F ⋙ shiftFunctor D b
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
f₂ : shiftFunctor D b ⋙ G ≅ G ⋙ shiftFunctor C b
h : CompatibilityUnit adj e₁ e₂
h' : CompatibilityUnit adj f₁ f₂
X : C
this :
  (shiftFunctor C b).map (adj.unit.app ((shiftFunctor C a).obj X)) ≫ f₂.inv.app (F.obj ((shiftFunctor C a).obj X)) =
    adj.unit.app ((shiftFunctor C b).obj ((shiftFunctor C a).obj X)) ≫ G.map (f₁.hom.app ((shiftFunctor C a).obj X))
| G.map (f₁.hom.app ((shiftFunctor C a).obj X))

case a.a.a.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
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a b : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
f₁ : shiftFunctor C b ⋙ F ≅ F ⋙ shiftFunctor D b
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
f₂ : shiftFunctor D b ⋙ G ≅ G ⋙ shiftFunctor C b
h : CompatibilityUnit adj e₁ e₂
h' : CompatibilityUnit adj f₁ f₂
X : C
this :
  (shiftFunctor C b).map (adj.unit.app ((shiftFunctor C a).obj X)) ≫ f₂.inv.app (F.obj ((shiftFunctor C a).obj X)) =
    adj.unit.app ((shiftFunctor C b).obj ((shiftFunctor C a).obj X)) ≫ G.map (f₁.hom.app ((shiftFunctor C a).obj X))
| G.map ((shiftFunctor D b).map (e₁.hom.app X))