Theorem: CategoryTheory.IsInitial.isVanKampenColimit
File: Mathlib/CategoryTheory/Limits/VanKampen.lean

tactic: intro F' c' α f hf hα
state_before: J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
K : Type u_1
inst✝² : Category.{?u.21664, u_1} K
D : Type u_2
inst✝¹ : Category.{?u.21671, u_2} D
inst✝ : HasStrictInitialObjects C
X : C
h : IsInitial X
⊢ IsVanKampenColimit (asEmptyCocone X)
state_after: J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
K : Type u_1
inst✝² : Category.{?u.21664, u_1} K
D : Type u_2
inst✝¹ : Category.{?u.21671, u_2} D
inst✝ : HasStrictInitialObjects C
X : C
h : IsInitial X
F' : Discrete PEmpty.{1} ⥤ C
c' : Cocone F'
α : F' ⟶ Functor.empty C
f : c'.pt ⟶ (asEmptyCocone X).pt
hf : α ≫ (asEmptyCocone X).ι = c'.ι ≫ (Functor.const (Discrete PEmpty.{1})).map f
hα : NatTrans.Equifibered α
⊢ Nonempty (IsColimit c') ↔ ∀ (j : Discrete PEmpty.{1}), IsPullback (c'.ι.app j) (α.app j) f ((asEmptyCocone X).ι.app j)

tactic: have : F' = Functor.empty C := by apply Functor.hext <;> rintro ⟨⟨⟩⟩
state_before: J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
K : Type u_1
inst✝² : Category.{?u.21664, u_1} K
D : Type u_2
inst✝¹ : Category.{?u.21671, u_2} D
inst✝ : HasStrictInitialObjects C
X : C
h : IsInitial X
F' : Discrete PEmpty.{1} ⥤ C
c' : Cocone F'
α : F' ⟶ Functor.empty C
f : c'.pt ⟶ (asEmptyCocone X).pt
hf : α ≫ (asEmptyCocone X).ι = c'.ι ≫ (Functor.const (Discrete PEmpty.{1})).map f
hα : NatTrans.Equifibered α
⊢ Nonempty (IsColimit c') ↔ ∀ (j : Discrete PEmpty.{1}), IsPullback (c'.ι.app j) (α.app j) f ((asEmptyCocone X).ι.app j)
state_after: J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
K : Type u_1
inst✝² : Category.{?u.21664, u_1} K
D : Type u_2
inst✝¹ : Category.{?u.21671, u_2} D
inst✝ : HasStrictInitialObjects C
X : C
h : IsInitial X
F' : Discrete PEmpty.{1} ⥤ C
c' : Cocone F'
α : F' ⟶ Functor.empty C
f : c'.pt ⟶ (asEmptyCocone X).pt
hf : α ≫ (asEmptyCocone X).ι = c'.ι ≫ (Functor.const (Discrete PEmpty.{1})).map f
hα : NatTrans.Equifibered α
this : F' = Functor.empty C
⊢ Nonempty (IsColimit c') ↔ ∀ (j : Discrete PEmpty.{1}), IsPullback (c'.ι.app j) (α.app j) f ((asEmptyCocone X).ι.app j)

tactic: subst this
state_before: J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
K : Type u_1
inst✝² : Category.{?u.21664, u_1} K
D : Type u_2
inst✝¹ : Category.{?u.21671, u_2} D
inst✝ : HasStrictInitialObjects C
X : C
h : IsInitial X
F' : Discrete PEmpty.{1} ⥤ C
c' : Cocone F'
α : F' ⟶ Functor.empty C
f : c'.pt ⟶ (asEmptyCocone X).pt
hf : α ≫ (asEmptyCocone X).ι = c'.ι ≫ (Functor.const (Discrete PEmpty.{1})).map f
hα : NatTrans.Equifibered α
this : F' = Functor.empty C
⊢ Nonempty (IsColimit c') ↔ ∀ (j : Discrete PEmpty.{1}), IsPullback (c'.ι.app j) (α.app j) f ((asEmptyCocone X).ι.app j)
state_after: J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
K : Type u_1
inst✝² : Category.{?u.21664, u_1} K
D : Type u_2
inst✝¹ : Category.{?u.21671, u_2} D
inst✝ : HasStrictInitialObjects C
X : C
h : IsInitial X
c' : Cocone (Functor.empty C)
α : Functor.empty C ⟶ Functor.empty C
f : c'.pt ⟶ (asEmptyCocone X).pt
hf : α ≫ (asEmptyCocone X).ι = c'.ι ≫ (Functor.const (Discrete PEmpty.{1})).map f
hα : NatTrans.Equifibered α
⊢ Nonempty (IsColimit c') ↔ ∀ (j : Discrete PEmpty.{1}), IsPullback (c'.ι.app j) (α.app j) f ((asEmptyCocone X).ι.app j)

tactic: haveI := h.isIso_to f
state_before: J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
K : Type u_1
inst✝² : Category.{?u.21664, u_1} K
D : Type u_2
inst✝¹ : Category.{?u.21671, u_2} D
inst✝ : HasStrictInitialObjects C
X : C
h : IsInitial X
c' : Cocone (Functor.empty C)
α : Functor.empty C ⟶ Functor.empty C
f : c'.pt ⟶ (asEmptyCocone X).pt
hf : α ≫ (asEmptyCocone X).ι = c'.ι ≫ (Functor.const (Discrete PEmpty.{1})).map f
hα : NatTrans.Equifibered α
⊢ Nonempty (IsColimit c') ↔ ∀ (j : Discrete PEmpty.{1}), IsPullback (c'.ι.app j) (α.app j) f ((asEmptyCocone X).ι.app j)
state_after: J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
K : Type u_1
inst✝² : Category.{?u.21664, u_1} K
D : Type u_2
inst✝¹ : Category.{?u.21671, u_2} D
inst✝ : HasStrictInitialObjects C
X : C
h : IsInitial X
c' : Cocone (Functor.empty C)
α : Functor.empty C ⟶ Functor.empty C
f : c'.pt ⟶ (asEmptyCocone X).pt
hf : α ≫ (asEmptyCocone X).ι = c'.ι ≫ (Functor.const (Discrete PEmpty.{1})).map f
hα : NatTrans.Equifibered α
this : IsIso f
⊢ Nonempty (IsColimit c') ↔ ∀ (j : Discrete PEmpty.{1}), IsPullback (c'.ι.app j) (α.app j) f ((asEmptyCocone X).ι.app j)

tactic: refine ⟨by rintro _ ⟨⟨⟩⟩,
  fun _ => ⟨IsColimit.ofIsoColimit h (Cocones.ext (asIso f).symm <| by rintro ⟨⟨⟩⟩)⟩⟩
state_before: J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
K : Type u_1
inst✝² : Category.{?u.21664, u_1} K
D : Type u_2
inst✝¹ : Category.{?u.21671, u_2} D
inst✝ : HasStrictInitialObjects C
X : C
h : IsInitial X
c' : Cocone (Functor.empty C)
α : Functor.empty C ⟶ Functor.empty C
f : c'.pt ⟶ (asEmptyCocone X).pt
hf : α ≫ (asEmptyCocone X).ι = c'.ι ≫ (Functor.const (Discrete PEmpty.{1})).map f
hα : NatTrans.Equifibered α
this : IsIso f
⊢ Nonempty (IsColimit c') ↔ ∀ (j : Discrete PEmpty.{1}), IsPullback (c'.ι.app j) (α.app j) f ((asEmptyCocone X).ι.app j)
state_after: no goals

tactic: apply Functor.hext <;> rintro ⟨⟨⟩⟩
state_before: J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
K : Type u_1
inst✝² : Category.{?u.21664, u_1} K
D : Type u_2
inst✝¹ : Category.{?u.21671, u_2} D
inst✝ : HasStrictInitialObjects C
X : C
h : IsInitial X
F' : Discrete PEmpty.{1} ⥤ C
c' : Cocone F'
α : F' ⟶ Functor.empty C
f : c'.pt ⟶ (asEmptyCocone X).pt
hf : α ≫ (asEmptyCocone X).ι = c'.ι ≫ (Functor.const (Discrete PEmpty.{1})).map f
hα : NatTrans.Equifibered α
⊢ F' = Functor.empty C
state_after: no goals

tactic: rintro _ ⟨⟨⟩⟩
state_before: J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
K : Type u_1
inst✝² : Category.{?u.21664, u_1} K
D : Type u_2
inst✝¹ : Category.{?u.21671, u_2} D
inst✝ : HasStrictInitialObjects C
X : C
h : IsInitial X
c' : Cocone (Functor.empty C)
α : Functor.empty C ⟶ Functor.empty C
f : c'.pt ⟶ (asEmptyCocone X).pt
hf : α ≫ (asEmptyCocone X).ι = c'.ι ≫ (Functor.const (Discrete PEmpty.{1})).map f
hα : NatTrans.Equifibered α
this : IsIso f
⊢ Nonempty (IsColimit c') → ∀ (j : Discrete PEmpty.{1}), IsPullback (c'.ι.app j) (α.app j) f ((asEmptyCocone X).ι.app j)
state_after: no goals

tactic: rintro ⟨⟨⟩⟩
state_before: J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
K : Type u_1
inst✝² : Category.{?u.21664, u_1} K
D : Type u_2
inst✝¹ : Category.{?u.21671, u_2} D
inst✝ : HasStrictInitialObjects C
X : C
h : IsInitial X
c' : Cocone (Functor.empty C)
α : Functor.empty C ⟶ Functor.empty C
f : c'.pt ⟶ (asEmptyCocone X).pt
hf : α ≫ (asEmptyCocone X).ι = c'.ι ≫ (Functor.const (Discrete PEmpty.{1})).map f
hα : NatTrans.Equifibered α
this : IsIso f
x✝ : ∀ (j : Discrete PEmpty.{1}), IsPullback (c'.ι.app j) (α.app j) f ((asEmptyCocone X).ι.app j)
⊢ ∀ (j : Discrete PEmpty.{1}), (asEmptyCocone X).ι.app j ≫ (asIso f).symm.hom = c'.ι.app j
state_after: no goals