-- 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 : (MultispanShape.prod ι).R)
      (x₁ : (c.multispanIndex (x := A) (u := U) (v := V).map Set.functorToTypes).right i₁)
      (x₂ : (c.multispanIndex (x := A) (u := U) (v := V).map Set.functorToTypes).right i₂),
      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) x₁)
          (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) x₂) →
        (c.multispanIndex (x := A) (u := U) (v := V).map Set.functorToTypes).multispan.ιColimitType
            (WalkingMultispan.right i₁) x₁ =
          (c.multispanIndex (x := A) (u := U) (v := V).map Set.functorToTypes).multispan.ιColimitType
            (WalkingMultispan.right i₂) x₂) ∧
    ∀
      (x :
        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))),
      ∃ (i : (MultispanShape.prod ι).R) (a :
        (c.multispanIndex (x := A) (u := U) (v := V).map Set.functorToTypes).right i),
        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 i) a =
          x

Tactic:
⟨fun i₁ i₂ ⟨x₁, h₁⟩ ⟨x₂, h₂⟩ h ↦ ?_, fun ⟨x, hx⟩ ↦ ?_⟩

Premises used: CompleteLattice.MulticoequalizerDiagram.multispanIndex, CategoryTheory.Limits.Types.isColimitOfMulticoequalizerDiagram.match_2, Exists, CategoryTheory.Limits.MultispanIndex.map, CategoryTheory.Limits.Multicofork.map, CategoryTheory.Functor.CoconeTypes.ι, CategoryTheory.Limits.WalkingMultispan.right, Subtype.mk, CategoryTheory.Limits.Types.isColimitOfMulticoequalizerDiagram.match_1, CategoryTheory.Functor.coconeTypesEquiv, CategoryTheory.Limits.MultispanIndex.multispan, CompleteLattice.MulticoequalizerDiagram.multicofork, And.intro, CategoryTheory.Limits.Types.isColimitOfMulticoequalizerDiagram.match_3, Set.functorToTypes, Equiv.symm, Eq, DFunLike.coe, CategoryTheory.Functor.ιColimitType