-- Module: Mathlib.Condensed.Discrete.Colimit
Goal:
X : Profiniteᵒᵖ ⥤ Type (u + 1)
inst : PreservesFiniteProducts X
hX : (S : Profinite.{u}) → IsColimit (X.mapCocone S.asLimitCone.op)
S : Profiniteᵒᵖ
Y : FintypeCatᵒᵖ
right : Discrete PUnit.{1}
g : toProfinite.op.obj Y ⟶ obj (fromPUnit S) right
f :
  LocallyConstant
    (↑((toProfinite.obj (Opposite.unop Y)).toTop (P := fun (X : TopCat.{u}) ↦
          TotallyDisconnectedSpace.{u} (↑X : Type u))) :
      Type u)
    (X.obj
      (Opposite.op (CompHausLike.of (fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) PUnit.{u + 1})))
x :
  Function.Fiber (Y :=
    (↑((toProfinite.obj (Opposite.unop Y)).toTop (P := fun (X : TopCat.{u}) ↦
          TotallyDisconnectedSpace.{u} (↑X : Type u))) :
      Type u))
    (Z :=
    X.obj
      (Opposite.op (CompHausLike.of (fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) PUnit.{u + 1})))
    (⇑f :
      (a : (↑(toProfinite.obj (Opposite.unop Y)).toTop : Type u)) →
        X.obj (Opposite.op (CompHausLike.of (fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u)) PUnit.{u + 1})))
this :
  Finite.{u + 1}
    (↑((fiber (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) f x).toTop (P :=
          fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
      Type u)
y :
  (↑((fiber (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) f x).toTop (P :=
        fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
    Type u)
w :
  Eq (α := X.obj (Opposite.op (Profinite.of PUnit.{u + 1})))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (X.map
              (CompHausLike.const (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))
                    (Profinite.of PUnit.{u + 1}) y ≫
                  sigmaIncl (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) f x).op) :
          (a : X.obj (Opposite.op (Profinite.of (Opposite.unop Y).obj))) →
            X.obj (Opposite.op (Profinite.of PUnit.{u + 1})))
        (((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
                (Iso.inv (isoFinYonedaComponents X (Profinite.of (Opposite.unop Y).obj))) :
              TypeCat.Fun ((Opposite.unop Y).obj → X.obj (Opposite.op (Profinite.of PUnit.{u + 1})))
                (X.obj (Opposite.op (Profinite.of (Opposite.unop Y).obj)))) :
            (a : (Opposite.unop Y).obj → X.obj (Opposite.op (Profinite.of PUnit.{u + 1}))) →
              X.obj (Opposite.op (Profinite.of (Opposite.unop Y).obj)))
          (⇑f : (a : (Opposite.unop Y).obj) → X.obj (Opposite.op (Profinite.of PUnit.{u + 1})))) :
      X.obj (Opposite.op (Profinite.of PUnit.{u + 1})))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (X.map
              (Quiver.Hom.op
                (CompHausLike.const (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))
                    (Profinite.of PUnit.{u + 1}) y ≫
                  IsTerminal.from
                    (CompHausLike.isTerminalPUnit (P := fun (X : TopCat.{u}) ↦
                      TotallyDisconnectedSpace.{u} (↑X : Type u)))
                    (fiber (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) f x)))) :
          (a :
              X.obj
                (Opposite.op
                  (CompHausLike.of (fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u)) PUnit.{u + 1}))) →
            X.obj (Opposite.op (Profinite.of PUnit.{u + 1})))
        (Function.Fiber.image
          (⇑f :
            (a : (Opposite.unop Y).obj) →
              X.obj
                (Opposite.op
                  (CompHausLike.of (fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u)) PUnit.{u + 1})))
          x) :
      X.obj (Opposite.op (Profinite.of PUnit.{u + 1})))
⊢ Eq (α := X.obj (Opposite.op (Profinite.of PUnit.{u + 1})))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (Iso.hom
              (isoFinYonedaComponents X
                (fiber (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) f x))) :
          (a : (fun (X : Type (u + 1)) ↦ X) (X.obj (Opposite.op (fiber f x)))) →
            (fun (X : Type (u + 1)) ↦ X)
              ((↑(fiber f x).toTop : Type u) → X.obj (Opposite.op (Profinite.of PUnit.{u + 1}))))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (X.map
                  (Quiver.Hom.op
                    (sigmaIncl (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) f x))) :
              (a : X.obj (Opposite.op (Profinite.of (Opposite.unop Y).obj))) → X.obj (Opposite.op (fiber f x)))
            (((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
                    (Iso.inv (isoFinYonedaComponents X (Profinite.of (Opposite.unop Y).obj))) :
                  TypeCat.Fun ((Opposite.unop Y).obj → X.obj (Opposite.op (Profinite.of PUnit.{u + 1})))
                    (X.obj (Opposite.op (Profinite.of (Opposite.unop Y).obj)))) :
                (a : (Opposite.unop Y).obj → X.obj (Opposite.op (Profinite.of PUnit.{u + 1}))) →
                  X.obj (Opposite.op (Profinite.of (Opposite.unop Y).obj)))
              (⇑f : (a : (Opposite.unop Y).obj) → X.obj (Opposite.op (Profinite.of PUnit.{u + 1})))) :
          X.obj (Opposite.op (fiber (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) f x)))
        y :
      X.obj (Opposite.op (Profinite.of PUnit.{u + 1})))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (Iso.hom
              (isoFinYonedaComponents X
                (fiber (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) f x))) :
          (a : (fun (X : Type (u + 1)) ↦ X) (X.obj (Opposite.op (fiber f x)))) →
            (fun (X : Type (u + 1)) ↦ X)
              ((↑(fiber f x).toTop : Type u) → X.obj (Opposite.op (Profinite.of PUnit.{u + 1}))))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (X.map
                  (Quiver.Hom.op
                    (IsTerminal.from
                      (CompHausLike.isTerminalPUnit (P := fun (X : TopCat.{u}) ↦
                        TotallyDisconnectedSpace.{u} (↑X : Type u)))
                      (fiber (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) f x)))) :
              (a :
                  X.obj
                    (Opposite.op
                      (CompHausLike.of (fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u)) PUnit.{u + 1}))) →
                X.obj (Opposite.op (fiber f x)))
            (Function.Fiber.image
              (⇑f :
                (a : (Opposite.unop Y).obj) →
                  X.obj
                    (Opposite.op
                      (CompHausLike.of (fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u)) PUnit.{u + 1})))
              x) :
          X.obj (Opposite.op (fiber (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) f x)))
        y :
      X.obj (Opposite.op (Profinite.of PUnit.{u + 1})))

Tactic:
simp only [toProfinite_obj, isoFinYonedaComponents_hom, ConcreteCategory.hom_ofHom,
    TypeCat.Fun.coe_mk, ← Functor.map_comp_apply, ← op_comp]

Premises used: CategoryTheory.op_comp, Condensed.isoFinYonedaComponents_hom, TypeCat.Fun.coe_mk, CategoryTheory.Functor.map_comp_apply, FintypeCat.toProfinite_obj, CategoryTheory.ConcreteCategory.hom_ofHom