Theorem: CategoryTheory.effectiveEpiFamilyStructOfEffectiveEpiDesc_aux
File: Mathlib/CategoryTheory/EffectiveEpi/Coproduct.lean

Proof steps:
state_before: C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π
⊢ (fun x => (Sigma.desc fun a => pullback.fst g₁ (Sigma.ι X a)) ≫ x) (g₁ ≫ Sigma.desc e) =
    (fun x => (Sigma.desc fun a => pullback.fst g₁ (Sigma.ι X a)) ≫ x) (g₂ ≫ Sigma.desc e)
state_after: case h
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π
a : α
⊢ Sigma.ι (fun a => pullback g₁ (Sigma.ι X a)) a ≫
      (fun x => (Sigma.desc fun a => pullback.fst g₁ (Sigma.ι X a)) ≫ x) (g₁ ≫ Sigma.desc e) =
    Sigma.ι (fun a => pullback g₁ (Sigma.ι X a)) a ≫
      (fun x => (Sigma.desc fun a => pullback.fst g₁ (Sigma.ι X a)) ≫ x) (g₂ ≫ Sigma.desc e)
tactic: ext a
state_before: case h
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π
a : α
⊢ Sigma.ι (fun a => pullback g₁ (Sigma.ι X a)) a ≫
      (fun x => (Sigma.desc fun a => pullback.fst g₁ (Sigma.ι X a)) ≫ x) (g₁ ≫ Sigma.desc e) =
    Sigma.ι (fun a => pullback g₁ (Sigma.ι X a)) a ≫
      (fun x => (Sigma.desc fun a => pullback.fst g₁ (Sigma.ι X a)) ≫ x) (g₂ ≫ Sigma.desc e)
state_after: case h
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π
a : α
⊢ pullback.fst g₁ (Sigma.ι X a) ≫ g₁ ≫ Sigma.desc e = pullback.fst g₁ (Sigma.ι X a) ≫ g₂ ≫ Sigma.desc e
tactic: simp only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_pt, Cofan.mk_ι_app]
state_before: case h
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π
a : α
⊢ pullback.fst g₁ (Sigma.ι X a) ≫ g₁ ≫ Sigma.desc e = pullback.fst g₁ (Sigma.ι X a) ≫ g₂ ≫ Sigma.desc e
state_after: case h
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π
a : α
⊢ (pullback.snd g₁ (Sigma.ι X a) ≫ Sigma.ι X a) ≫ Sigma.desc e = pullback.fst g₁ (Sigma.ι X a) ≫ g₂ ≫ Sigma.desc e
tactic: rw [← Category.assoc, pullback.condition]
state_before: case h
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π
a : α
⊢ (pullback.snd g₁ (Sigma.ι X a) ≫ Sigma.ι X a) ≫ Sigma.desc e = pullback.fst g₁ (Sigma.ι X a) ≫ g₂ ≫ Sigma.desc e
state_after: case h
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π
a : α
⊢ pullback.snd g₁ (Sigma.ι X a) ≫ e a = pullback.fst g₁ (Sigma.ι X a) ≫ g₂ ≫ Sigma.desc e
tactic: simp only [Category.assoc, colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app]
state_before: case h
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π
a : α
⊢ (fun x => (Sigma.desc fun a_1 => pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X a_1)) ≫ x)
      (pullback.snd g₁ (Sigma.ι X a) ≫ e a) =
    (fun x => (Sigma.desc fun a_1 => pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X a_1)) ≫ x)
      (pullback.fst g₁ (Sigma.ι X a) ≫ g₂ ≫ Sigma.desc e)
state_after: case h.h
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π
a b : α
⊢ Sigma.ι (fun a_1 => pullback (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X a_1)) b ≫
      (fun x => (Sigma.desc fun a_1 => pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X a_1)) ≫ x)
        (pullback.snd g₁ (Sigma.ι X a) ≫ e a) =
    Sigma.ι (fun a_1 => pullback (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X a_1)) b ≫
      (fun x => (Sigma.desc fun a_1 => pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X a_1)) ≫ x)
        (pullback.fst g₁ (Sigma.ι X a) ≫ g₂ ≫ Sigma.desc e)
tactic: ext b
state_before: case h.h
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π
a b : α
⊢ Sigma.ι (fun a_1 => pullback (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X a_1)) b ≫
      (fun x => (Sigma.desc fun a_1 => pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X a_1)) ≫ x)
        (pullback.snd g₁ (Sigma.ι X a) ≫ e a) =
    Sigma.ι (fun a_1 => pullback (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X a_1)) b ≫
      (fun x => (Sigma.desc fun a_1 => pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X a_1)) ≫ x)
        (pullback.fst g₁ (Sigma.ι X a) ≫ g₂ ≫ Sigma.desc e)
state_after: case h.h
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π
a b : α
⊢ pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a) ≫ e a =
    pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.fst g₁ (Sigma.ι X a) ≫ g₂ ≫ Sigma.desc e
tactic: simp only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_pt, Cofan.mk_ι_app]
state_before: case h.h
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π
a b : α
⊢ pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a) ≫ e a =
    pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.fst g₁ (Sigma.ι X a) ≫ g₂ ≫ Sigma.desc e
state_after: case h.h
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π
a b : α
⊢ (pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a)) ≫ e a =
    ((pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.fst g₁ (Sigma.ι X a)) ≫ g₂) ≫
      Sigma.desc e
tactic: simp only [← Category.assoc]
state_before: case h.h
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π
a b : α
⊢ (pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a)) ≫ e a =
    ((pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.fst g₁ (Sigma.ι X a)) ≫ g₂) ≫
      Sigma.desc e
state_after: case h.h
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π
a b : α
⊢ (pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a)) ≫ e a =
    (pullback.snd (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ Sigma.ι X b) ≫ Sigma.desc e
tactic: rw [(Category.assoc _ _ g₂), pullback.condition]
state_before: case h.h
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π
a b : α
⊢ (pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a)) ≫ e a =
    (pullback.snd (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ Sigma.ι X b) ≫ Sigma.desc e
state_after: case h.h
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π
a b : α
⊢ pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a) ≫ e a =
    pullback.snd (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ e b
tactic: simp only [Category.assoc, colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app]
state_before: case h.h
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π
a b : α
⊢ pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a) ≫ e a =
    pullback.snd (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ e b
state_after: case h.h
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π
a b : α
⊢ (pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a)) ≫ e a =
    pullback.snd (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ e b
tactic: rw [← Category.assoc]
state_before: case h.h
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π
a b : α
⊢ (pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a)) ≫ e a =
    pullback.snd (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ e b
state_after: case h.h.a
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π
a b : α
⊢ (pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a)) ≫ π a =
    pullback.snd (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ π b
tactic: apply h
state_before: case h.h.a
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
a b : α
hg : pullback.fst g₁ (Sigma.ι X a) ≫ g₁ ≫ Sigma.desc π = pullback.fst g₁ (Sigma.ι X a) ≫ g₂ ≫ Sigma.desc π
⊢ (pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a)) ≫ π a =
    pullback.snd (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ π b
state_after: case h.h.a
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
a b : α
hg : (pullback.snd g₁ (Sigma.ι X a) ≫ Sigma.ι X a) ≫ Sigma.desc π = pullback.fst g₁ (Sigma.ι X a) ≫ g₂ ≫ Sigma.desc π
⊢ (pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a)) ≫ π a =
    pullback.snd (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ π b
tactic: rw [← Category.assoc, pullback.condition] at hg
state_before: case h.h.a
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
a b : α
hg : (pullback.snd g₁ (Sigma.ι X a) ≫ Sigma.ι X a) ≫ Sigma.desc π = pullback.fst g₁ (Sigma.ι X a) ≫ g₂ ≫ Sigma.desc π
⊢ (pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a)) ≫ π a =
    pullback.snd (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ π b
state_after: case h.h.a
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
a b : α
hg : pullback.snd g₁ (Sigma.ι X a) ≫ π a = pullback.fst g₁ (Sigma.ι X a) ≫ g₂ ≫ Sigma.desc π
⊢ (pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a)) ≫ π a =
    pullback.snd (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ π b
tactic: simp only [Category.assoc, colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app] at hg
state_before: case h.h.a
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
a b : α
hg :
  Sigma.ι (fun a_1 => pullback (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X a_1)) b ≫
      (Sigma.desc fun a_1 => pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X a_1)) ≫
        pullback.snd g₁ (Sigma.ι X a) ≫ π a =
    Sigma.ι (fun a_1 => pullback (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X a_1)) b ≫
      (Sigma.desc fun a_1 => pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X a_1)) ≫
        pullback.fst g₁ (Sigma.ι X a) ≫ g₂ ≫ Sigma.desc π
⊢ (pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a)) ≫ π a =
    pullback.snd (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ π b
state_after: case h.h.a
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
a b : α
hg :
  pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a) ≫ π a =
    pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.fst g₁ (Sigma.ι X a) ≫ g₂ ≫ Sigma.desc π
⊢ (pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a)) ≫ π a =
    pullback.snd (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ π b
tactic: simp only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_pt, Cofan.mk_ι_app] at hg
state_before: case h.h.a
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
a b : α
hg :
  pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a) ≫ π a =
    pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.fst g₁ (Sigma.ι X a) ≫ g₂ ≫ Sigma.desc π
⊢ (pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a)) ≫ π a =
    pullback.snd (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ π b
state_after: case h.h.a
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
a b : α
hg :
  (pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a)) ≫ π a =
    ((pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.fst g₁ (Sigma.ι X a)) ≫ g₂) ≫
      Sigma.desc π
⊢ (pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a)) ≫ π a =
    pullback.snd (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ π b
tactic: simp only [← Category.assoc] at hg
state_before: case h.h.a
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
a b : α
hg :
  (pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a)) ≫ π a =
    ((pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.fst g₁ (Sigma.ι X a)) ≫ g₂) ≫
      Sigma.desc π
⊢ (pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a)) ≫ π a =
    pullback.snd (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ π b
state_after: case h.h.a
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
a b : α
hg :
  (pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a)) ≫ π a =
    (pullback.snd (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ Sigma.ι X b) ≫ Sigma.desc π
⊢ (pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a)) ≫ π a =
    pullback.snd (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ π b
tactic: rw [(Category.assoc _ _ g₂), pullback.condition] at hg
state_before: case h.h.a
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
B : C
α : Type u_2
X : α → C
π : (a : α) → X a ⟶ B
inst✝³ : HasCoproduct X
inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)
inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)
inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
Z : C
g₁ g₂ : Z ⟶ ∐ fun b => X b
a b : α
hg :
  (pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a)) ≫ π a =
    (pullback.snd (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ Sigma.ι X b) ≫ Sigma.desc π
⊢ (pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a)) ≫ π a =
    pullback.snd (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ π b
state_after: no goals
tactic: simpa using hg