-- Module: Mathlib.Condensed.Light.InternallyProjective
Goal:
R : Type u
inst : CommRing R
P : LightProfinite
rw :
  (∀ {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)))
              P.toCondensed 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' :
        obj (free R)
            (tensorObj (𝒞 :=
              ObjectProperty.FullSubcategory.category (Presheaf.IsSheaf (coherentTopology LightProfinite)))
              P.toCondensed S'.toCondensed) ⟶
          A),
        CategoryTheory.Functor.map (free R) (P.toCondensed ◁ lightProfiniteToLightCondSet.map π) ≫ g = g' ≫ e) ↔
    ∀ {A B : LightCondMod R} (e : A ⟶ B) [Epi e] (S : LightProfinite.{u})
      (g :
        obj (free R)
            (tensorObj (𝒞 :=
                CompHausLike.category fun (X : TopCat.{u}) ↦
                  TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))
                P 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' :
        obj (free R)
            (tensorObj (𝒞 :=
                CompHausLike.category fun (X : TopCat.{u}) ↦
                  TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))
                P S').toCondensed ⟶
          A),
        CategoryTheory.Functor.map (free R) (lightProfiniteToLightCondSet.map (P ◁ π)) ≫ g = g' ≫ e
⊢ InternallyProjective (C := LightCondMod R) (obj (free R) P.toCondensed) ↔
    ∀ {A B : LightCondMod R} (e : A ⟶ B) [Epi e] (S : LightProfinite.{u})
      (g :
        obj (free R)
            (tensorObj (𝒞 :=
                CompHausLike.category fun (X : TopCat.{u}) ↦
                  TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))
                P 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' :
        obj (free R)
            (tensorObj (𝒞 :=
                CompHausLike.category fun (X : TopCat.{u}) ↦
                  TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))
                P S').toCondensed ⟶
          A),
        CategoryTheory.Functor.map (free R) (lightProfiniteToLightCondSet.map (P ◁ π)) ≫ g = g' ≫ e

Tactic:
rw [free_internallyProjective_iff_tensor_condition]

Premises used: LightCondensed.free_internallyProjective_iff_tensor_condition