-- Module: Mathlib.Topology.Category.CompHausLike.Limits
Goal:
P : TopCat → Prop
α : Type w
inst : Finite α
X : α → CompHausLike P
inst_1 : HasExplicitFiniteCoproduct.{w, u} (P := P) X
a : α
e_2 :
  (↑((finiteCoproduct.cofan.{w, u} (P := P) X).pt.toTop (P := P)) : Type (max u w)) =
    (fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type (max u w))) (finiteCoproduct.{w, u} (P := P) X)
h :
  ∀ (x : (fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type (max u w))) (X a)),
    Eq (α := (↑((finiteCoproduct.cofan.{w, u} (P := P) X).pt.toTop (P := P)) : Type (max u w)))
      (Function.comp (β := (fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type (max u w))) (∐ X))
          (⇑(homeoOfIso (P := P)
                (IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X))
                  (finiteCoproduct.isColimit (P := P) X))) :
            (a : (↑(colimit.cocone (Discrete.functor X)).pt.toTop : Type (max u w))) →
              (↑(finiteCoproduct.cofan X).pt.toTop : Type (max u w)))
          (⇑(ConcreteCategory.hom (FC := fun (x1 x2 : CompHausLike P) ↦
                C((↑(x1.toTop (P := P)) : Type (max u w)), (↑(x2.toTop (P := P)) : Type (max u w)))) (CC :=
                fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type (max u w))) (Y := ∐ X) (Sigma.ι X a)) :
            (a : (fun (X : CompHausLike P) ↦ (↑X.toTop : Type (max u w))) (X a)) →
              (fun (X : CompHausLike P) ↦ (↑X.toTop : Type (max u w))) (∐ X))
          x :
        (↑((finiteCoproduct.cofan.{w, u} (P := P) X).pt.toTop (P := P)) : Type (max u w)))
      ((ConcreteCategory.hom (FC := fun (x1 x2 : CompHausLike P) ↦
              C((↑(x1.toTop (P := P)) : Type (max u w)), (↑(x2.toTop (P := P)) : Type (max u w)))) (CC :=
              fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type (max u w))) (Y :=
              finiteCoproduct.{w, u} (P := P) X) (finiteCoproduct.ι (P := P) X a) :
            (a : (fun (X : CompHausLike P) ↦ (↑X.toTop : Type (max u w))) (X a)) →
              (fun (X : CompHausLike P) ↦ (↑X.toTop : Type (max u w))) (finiteCoproduct X))
          x :
        (fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type (max u w))) (finiteCoproduct.{w, u} (P := P) X))
⊢ Eq (α :=
    (fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type (max u w))) (X a) →
      (↑((finiteCoproduct.cofan.{w, u} (P := P) X).pt.toTop (P := P)) : Type (max u w)))
    (Function.comp (β := (fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type (max u w))) (∐ X))
      (⇑(homeoOfIso (P := P)
            (IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X))
              (finiteCoproduct.isColimit (P := P) X))) :
        (a : (↑(colimit.cocone (Discrete.functor X)).pt.toTop : Type (max u w))) →
          (↑(finiteCoproduct.cofan X).pt.toTop : Type (max u w)))
      (⇑(ConcreteCategory.hom (FC := fun (x1 x2 : CompHausLike P) ↦
            C((↑(x1.toTop (P := P)) : Type (max u w)), (↑(x2.toTop (P := P)) : Type (max u w)))) (CC :=
            fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type (max u w))) (Y := ∐ X) (Sigma.ι X a)) :
        (a : (fun (X : CompHausLike P) ↦ (↑X.toTop : Type (max u w))) (X a)) →
          (fun (X : CompHausLike P) ↦ (↑X.toTop : Type (max u w))) (∐ X)))
    (⇑(ConcreteCategory.hom (FC := fun (x1 x2 : CompHausLike P) ↦
          C((↑(x1.toTop (P := P)) : Type (max u w)), (↑(x2.toTop (P := P)) : Type (max u w)))) (CC :=
          fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type (max u w))) (Y := finiteCoproduct.{w, u} (P := P) X)
          (finiteCoproduct.ι (P := P) X a)) :
      (a : (fun (X : CompHausLike P) ↦ (↑X.toTop : Type (max u w))) (X a)) →
        (fun (X : CompHausLike P) ↦ (↑X.toTop : Type (max u w))) (finiteCoproduct X))

Tactic:
ext x