-- Module: Mathlib.Condensed.Light.InternallyProjective
Goal:
R : Type u
inst : CommRing R
P : LightCondMod R
h :
  ∀ {A B : LightCondMod R} (e : A ⟶ B) [Epi e] (S : LightProfinite.{u})
    (g :
      tensorObj (𝒞 := ObjectProperty.FullSubcategory.category (Presheaf.IsSheaf (coherentTopology LightProfinite))) P
          (obj (free R) S.toCondensed) ⟶
        B),
    ∃ (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' :
      tensorObj (𝒞 := ObjectProperty.FullSubcategory.category (Presheaf.IsSheaf (coherentTopology LightProfinite))) P
          (obj (free R) S'.toCondensed) ⟶
        A),
      P ◁ CategoryTheory.Functor.map (lightProfiniteToLightCondSet ⋙ free R) π ≫ g = g' ≫ e
A B : LightCondMod R
e : A ⟶ B
he : Epi e
refine_2 :
  ∀ (S : LightProfinite.{u})
    (y : (↑((P ⟹ B).obj (P := Presheaf.IsSheaf (coherentTopology LightProfinite)).obj (Opposite.op S)) : Type u)),
    ∃ (S' : LightProfinite.{u}) (φ : 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))
              φ) :
          (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))
      (x : (↑((P ⟹ A).obj (P := Presheaf.IsSheaf (coherentTopology LightProfinite)).obj (Opposite.op S')) : Type u)),
      Eq (α := (↑((P ⟹ B).obj (P := Presheaf.IsSheaf (coherentTopology LightProfinite)).obj (Opposite.op S')) : Type u))
        ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type u) →ₗ[R] (↑x2 : Type u)) (CC :=
                ModuleCat.carrier) (X :=
                (P ⟹ A).obj (P := Presheaf.IsSheaf (coherentTopology LightProfinite)).obj (Opposite.op S'))
                (NatTrans.app (InducedCategory.Hom.hom (CategoryTheory.Functor.map (ihom P) e)) (Opposite.op S')) :
              (a : (↑((P ⟹ A).obj.obj (Opposite.op S')) : Type u)) → (↑((P ⟹ B).obj.obj (Opposite.op S')) : Type u))
            x :
          (↑((P ⟹ B).obj (P := Presheaf.IsSheaf (coherentTopology LightProfinite)).obj (Opposite.op S')) : Type u))
        ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type u) →ₗ[R] (↑x2 : Type u)) (CC :=
                ModuleCat.carrier)
                ((P ⟹ B).obj (P := Presheaf.IsSheaf (coherentTopology LightProfinite)).map (Opposite.op φ)) :
              (a : (↑((P ⟹ B).obj.obj (Opposite.op S)) : Type u)) → (↑((P ⟹ B).obj.obj (Opposite.op S')) : Type u))
            y :
          (↑((P ⟹ B).obj (P := Presheaf.IsSheaf (coherentTopology LightProfinite)).obj (Opposite.op S')) : Type u))
⊢ Epi (X := P ⟹ A) (CategoryTheory.Functor.map (ihom P) e)

Tactic:
rw [LightCondMod.epi_iff_locallySurjective_on_lightProfinite]

Premises used: LightCondMod.epi_iff_locallySurjective_on_lightProfinite