-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Types.Multicoequalizer (goal)
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
⊢ Nonempty (IsColimit (c.multicofork (x := A) (u := U) (v := V).map Set.functorToTypes))
have h : (∀ (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: rw [Types.isColimit_iff_coconeTypesIsColimit,
    Functor.CoconeTypes.isMulticoequalizer_iff]