-- Module: Mathlib.CategoryTheory.Limits.Types.Multicoequalizer
Goal:
X : Type u
ι : Type w
A : Set X
U : ι → Set X
V : ι → ι → Set X
c : MulticoequalizerDiagram A U V
e : Functor.{w, u, w, u + 1} (WalkingMultispan (MultispanShape.prod ι)) (Type u) :=
  (c.multispanIndex (x := A) (u := U) (v := V).map Set.functorToTypes).multispan
i₁ i₂ : ι
x : (c.multispanIndex (x := A) (u := U) (v := V).map Set.functorToTypes).right i₁
x_1 : (c.multispanIndex (x := A) (u := U) (v := V).map Set.functorToTypes).right i₂
x₁ : X
h₁ : x₁ ∈ U i₁
x₂ : X
h₂ : x₂ ∈ U i₂
h :
  Eq.{u + 1} (α :=
    Functor.CoconeTypes.pt.{u, u, w, w} (J := WalkingMultispan (MultispanShape.prod ι)) (F :=
      (c.multispanIndex (x := A) (u := U) (v := V).map Set.functorToTypes).multispan)
      (((c.multispanIndex (x := A) (u := U) (v := V).map Set.functorToTypes).multispan.coconeTypesEquiv.symm :
          (a : Cocone (c.multispanIndex.map Set.functorToTypes).multispan) →
            (c.multispanIndex.map Set.functorToTypes).multispan.CoconeTypes)
        (c.multicofork (x := A) (u := U) (v := V).map Set.functorToTypes)))
    (Functor.CoconeTypes.ι
      (((c.multispanIndex (x := A) (u := U) (v := V).map Set.functorToTypes).multispan.coconeTypesEquiv.symm :
          (a : Cocone (c.multispanIndex.map Set.functorToTypes).multispan) →
            (c.multispanIndex.map Set.functorToTypes).multispan.CoconeTypes)
        (c.multicofork (x := A) (u := U) (v := V).map Set.functorToTypes))
      (WalkingMultispan.right i₁)
      (Subtype.mk (p := fun (x : X) ↦ x ∈ c.multispanIndex (x := A) (u := U) (v := V).right i₁) x₁ h₁))
    (Functor.CoconeTypes.ι
      (((c.multispanIndex (x := A) (u := U) (v := V).map Set.functorToTypes).multispan.coconeTypesEquiv.symm :
          (a : Cocone (c.multispanIndex.map Set.functorToTypes).multispan) →
            (c.multispanIndex.map Set.functorToTypes).multispan.CoconeTypes)
        (c.multicofork (x := A) (u := U) (v := V).map Set.functorToTypes))
      (WalkingMultispan.right i₂)
      (Subtype.mk (p := fun (x : X) ↦ x ∈ c.multispanIndex (x := A) (u := U) (v := V).right i₂) x₂ h₂))
h₃ :
  ∀ (h₂ : x₁ ∈ U i₂)
    (h :
      Eq.{u + 1} (α :=
        Functor.CoconeTypes.pt.{u, u, w, w} (J := WalkingMultispan (MultispanShape.prod ι)) (F :=
          (c.multispanIndex (x := A) (u := U) (v := V).map Set.functorToTypes).multispan)
          (((c.multispanIndex (x := A) (u := U) (v := V).map Set.functorToTypes).multispan.coconeTypesEquiv.symm :
              (a : Cocone (c.multispanIndex.map Set.functorToTypes).multispan) →
                (c.multispanIndex.map Set.functorToTypes).multispan.CoconeTypes)
            (c.multicofork (x := A) (u := U) (v := V).map Set.functorToTypes)))
        (Functor.CoconeTypes.ι
          (((c.multispanIndex (x := A) (u := U) (v := V).map Set.functorToTypes).multispan.coconeTypesEquiv.symm :
              (a : Cocone (c.multispanIndex.map Set.functorToTypes).multispan) →
                (c.multispanIndex.map Set.functorToTypes).multispan.CoconeTypes)
            (c.multicofork (x := A) (u := U) (v := V).map Set.functorToTypes))
          (WalkingMultispan.right i₁)
          (Subtype.mk (p := fun (x : X) ↦ x ∈ c.multispanIndex (x := A) (u := U) (v := V).right i₁) x₁ h₁))
        (Functor.CoconeTypes.ι
          (((c.multispanIndex (x := A) (u := U) (v := V).map Set.functorToTypes).multispan.coconeTypesEquiv.symm :
              (a : Cocone (c.multispanIndex.map Set.functorToTypes).multispan) →
                (c.multispanIndex.map Set.functorToTypes).multispan.CoconeTypes)
            (c.multicofork (x := A) (u := U) (v := V).map Set.functorToTypes))
          (WalkingMultispan.right i₂)
          (Subtype.mk (p := fun (x : X) ↦ x ∈ c.multispanIndex (x := A) (u := U) (v := V).right i₂) x₁ h₂))),
    (c.multispanIndex (x := A) (u := U) (v := V).map Set.functorToTypes).multispan.ιColimitType
        (WalkingMultispan.right i₁)
        (Subtype.mk (p := fun (x : X) ↦ x ∈ c.multispanIndex (x := A) (u := U) (v := V).right i₁) x₁ h₁) =
      (c.multispanIndex (x := A) (u := U) (v := V).map Set.functorToTypes).multispan.ιColimitType
        (WalkingMultispan.right i₂)
        (Subtype.mk (p := fun (x : X) ↦ x ∈ c.multispanIndex (x := A) (u := U) (v := V).right i₂) x₁ h₂)
⊢ (c.multispanIndex (x := A) (u := U) (v := V).map Set.functorToTypes).multispan.ιColimitType
      (WalkingMultispan.right i₁)
      (Subtype.mk (p := fun (x : X) ↦ x ∈ c.multispanIndex (x := A) (u := U) (v := V).right i₁) x₁ h₁) =
    (c.multispanIndex (x := A) (u := U) (v := V).map Set.functorToTypes).multispan.ιColimitType
      (WalkingMultispan.right i₂)
      (Subtype.mk (p := fun (x : X) ↦ x ∈ c.multispanIndex (x := A) (u := U) (v := V).right i₂) x₂ h₂)

Tactic:
obtain rfl : x₁ = x₂ := by simpa using h

Premises used: rfl