-- 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)
⊢ (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).op) :
        (a : X.obj (Opposite.op (fiber f x))) → 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}))))) =
    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

Tactic:
simpa [← dsimp% isoFinYonedaComponents_inv_comp X _ (sigmaIncl.{u, u + 1} f x),
    ← isoFinYonedaComponents_hom_apply, -isoFinYonedaComponents_hom] using x.map_eq_image f y

Premises used: Function.Fiber.map_eq_image, Condensed.isoFinYonedaComponents_hom, Condensed.isoFinYonedaComponents_hom_apply, DFunLike.coe