-- Module: Mathlib.Condensed.Light.InternallyProjective
Goal:
R : Type u
inst : CommRing R
P : LightCondSet
refine_1 :
  ∀
    (h :
      ∀ {A B : LightCondMod R} (e : A ⟶ B) [Epi e] (S : LightProfinite.{u})
        (g :
          tensorObj (𝒞 := ObjectProperty.FullSubcategory.category (Presheaf.IsSheaf (coherentTopology LightProfinite)))
              (obj (free R) S.toCondensed) (obj (free R) P) ⟶
            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)))
              (obj (free R) S'.toCondensed) (obj (free R) P) ⟶
            A),
          CategoryTheory.Functor.map (lightProfiniteToLightCondSet ⋙ free R) π ▷ obj (free R) P ≫ g = g' ≫ e)
    (A B : LightCondMod R) (e : A ⟶ B) (he : Epi e) (S : LightProfinite.{u})
    (g :
      obj (free R)
          (tensorObj (𝒞 := ObjectProperty.FullSubcategory.category (Presheaf.IsSheaf (coherentTopology LightProfinite)))
            S.toCondensed P) ⟶
        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' :
      obj (free R)
          (tensorObj (𝒞 := ObjectProperty.FullSubcategory.category (Presheaf.IsSheaf (coherentTopology LightProfinite)))
            S'.toCondensed P) ⟶
        A),
      CategoryTheory.Functor.map (free R) (lightProfiniteToLightCondSet.map π ▷ P) ≫ g = g' ≫ e
refine_2 :
  ∀
    (h :
      ∀ {A B : LightCondMod R} (e : A ⟶ B) [Epi e] (S : LightProfinite.{u})
        (g :
          obj (free R)
              (tensorObj (𝒞 :=
                ObjectProperty.FullSubcategory.category (Presheaf.IsSheaf (coherentTopology LightProfinite)))
                S.toCondensed P) ⟶
            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' :
          obj (free R)
              (tensorObj (𝒞 :=
                ObjectProperty.FullSubcategory.category (Presheaf.IsSheaf (coherentTopology LightProfinite)))
                S'.toCondensed P) ⟶
            A),
          CategoryTheory.Functor.map (free R) (lightProfiniteToLightCondSet.map π ▷ P) ≫ g = g' ≫ e)
    (A B : LightCondMod R) (e : A ⟶ B) (he : Epi e) (S : LightProfinite.{u})
    (g :
      tensorObj (𝒞 := ObjectProperty.FullSubcategory.category (Presheaf.IsSheaf (coherentTopology LightProfinite)))
          (obj (free R) S.toCondensed) (obj (free R) P) ⟶
        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)))
          (obj (free R) S'.toCondensed) (obj (free R) P) ⟶
        A),
      CategoryTheory.Functor.map (lightProfiniteToLightCondSet ⋙ free R) π ▷ obj (free R) P ≫ g = g' ≫ e
⊢ (∀ {A B : LightCondMod R} (e : A ⟶ B) [Epi e] (S : LightProfinite.{u})
      (g :
        tensorObj (𝒞 := ObjectProperty.FullSubcategory.category (Presheaf.IsSheaf (coherentTopology LightProfinite)))
            (obj (free R) S.toCondensed) (obj (free R) P) ⟶
          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)))
            (obj (free R) S'.toCondensed) (obj (free R) P) ⟶
          A),
        CategoryTheory.Functor.map (lightProfiniteToLightCondSet ⋙ free R) π ▷ obj (free R) P ≫ g = g' ≫ e) ↔
    ∀ {A B : LightCondMod R} (e : A ⟶ B) [Epi e] (S : LightProfinite.{u})
      (g :
        obj (free R)
            (tensorObj (𝒞 :=
              ObjectProperty.FullSubcategory.category (Presheaf.IsSheaf (coherentTopology LightProfinite)))
              S.toCondensed P) ⟶
          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' :
        obj (free R)
            (tensorObj (𝒞 :=
              ObjectProperty.FullSubcategory.category (Presheaf.IsSheaf (coherentTopology LightProfinite)))
              S'.toCondensed P) ⟶
          A),
        CategoryTheory.Functor.map (free R) (lightProfiniteToLightCondSet.map π ▷ P) ≫ g = g' ≫ e

Tactic:
refine ⟨fun h A B e he S g ↦ ?_, fun h A B e he S g ↦ ?_⟩