-- Module: Mathlib.Condensed.Light.InternallyProjective
Goal:
R : Type u
inst : CommRing R
P : LightProfinite
A B : LightCondMod R
e : A ⟶ B
he : Epi e
S : LightProfinite
g :
  obj (free R)
      (tensorObj (𝒞 := ObjectProperty.FullSubcategory.category (Presheaf.IsSheaf (coherentTopology LightProfinite)))
        S.toCondensed P.toCondensed) ⟶
    B
S' : LightProfinite
π : S' ⟶ S
hπ :
  Function.Surjective (α :=
    (fun
          (X :
            CompHausLike fun (X : TopCat.{u}) ↦
              TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
        (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
              TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
          Type u))
      S')
    (β :=
    (fun
          (X :
            CompHausLike fun (X : TopCat.{u}) ↦
              TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
        (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
              TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
          Type u))
      S)
    (⇑(ConcreteCategory.hom (FC :=
          fun
            (x1 x2 :
              CompHausLike fun (X : TopCat.{u}) ↦
                TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
          C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦
                  TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
              Type u),
            (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦
                  TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
              Type u)))
          (CC :=
          fun
            (X :
              CompHausLike fun (X : TopCat.{u}) ↦
                TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
          (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
                TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
            Type u))
          π) :
      (a :
          (fun
                (X :
                  CompHausLike fun (X : TopCat) ↦
                    TotallyDisconnectedSpace (↑X : Type u) ∧ SecondCountableTopology (↑X : Type u)) ↦
              (↑X.toTop : Type u))
            S') →
        (fun
              (X :
                CompHausLike fun (X : TopCat) ↦
                  TotallyDisconnectedSpace (↑X : Type u) ∧ SecondCountableTopology (↑X : Type u)) ↦
            (↑X.toTop : Type u))
          S)
g' :
  obj (free R)
      (tensorObj (𝒞 :=
          CompHausLike.category fun (X : TopCat.{u}) ↦
            TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))
          S' P).toCondensed ⟶
    A
hh :
  CategoryTheory.Functor.map (free R) (lightProfiniteToLightCondSet.map (π ▷ P)) ≫
      CategoryTheory.Functor.map (free R) (Iso.inv (μIso lightProfiniteToLightCondSet S P)) ≫ g =
    g' ≫ e
refine_2 :
  CategoryTheory.Functor.map (free R) (lightProfiniteToLightCondSet.map π ▷ P.toCondensed) ≫ g =
    (CategoryTheory.Functor.map (free R) (Iso.hom (μIso lightProfiniteToLightCondSet S' P)) ≫ g') ≫ e
⊢ ∃ (S' : LightProfinite.{u}) (π : S' ⟶ S) (_ :
    Function.Surjective (α :=
      (fun
            (X :
              CompHausLike fun (X : TopCat.{u}) ↦
                TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
          (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
                TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
            Type u))
        S')
      (β :=
      (fun
            (X :
              CompHausLike fun (X : TopCat.{u}) ↦
                TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
          (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
                TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
            Type u))
        S)
      (⇑(ConcreteCategory.hom (FC :=
            fun
              (x1 x2 :
                CompHausLike fun (X : TopCat.{u}) ↦
                  TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
            C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦
                    TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
                Type u),
              (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦
                    TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
                Type u)))
            (CC :=
            fun
              (X :
                CompHausLike fun (X : TopCat.{u}) ↦
                  TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
            (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
                  TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
              Type u))
            π) :
        (a :
            (fun
                  (X :
                    CompHausLike fun (X : TopCat) ↦
                      TotallyDisconnectedSpace (↑X : Type u) ∧ SecondCountableTopology (↑X : Type u)) ↦
                (↑X.toTop : Type u))
              S') →
          (fun
                (X :
                  CompHausLike fun (X : TopCat) ↦
                    TotallyDisconnectedSpace (↑X : Type u) ∧ SecondCountableTopology (↑X : Type u)) ↦
              (↑X.toTop : Type u))
            S))
    (g' :
    obj (free R)
        (tensorObj (𝒞 := ObjectProperty.FullSubcategory.category (Presheaf.IsSheaf (coherentTopology LightProfinite)))
          S'.toCondensed P.toCondensed) ⟶
      A),
    CategoryTheory.Functor.map (free R) (lightProfiniteToLightCondSet.map π ▷ P.toCondensed) ≫ g = g' ≫ e

Tactic:
refine ⟨S', π, hπ, (free R).map
      (μIso lightProfiniteToLightCondSet _ _).hom ≫ g', ?_⟩