-- Module: Mathlib.Condensed.Light.Sequence
Goal:
R : Type
inst : CommRing R
X : LightCondMod R
S T : LightProfinite
π :
  T ⟶
    tensorObj (𝒞 :=
      CompHausLike.category fun (X : TopCat.{0}) ↦
        TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type))
      S ℕ∪{∞}
inst_1 :
  Epi (C := LightCondMod R) (X :=
    obj (lightProfiniteToLightCondSet ⋙ free R)
      (CompHausLike.of
        (fun (X : TopCat.{0}) ↦ TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type))
        ((↑(T.toTop (P := fun (X : TopCat.{0}) ↦
                TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type))) :
            Type) ⊕
          (↑((CompHausLike.pullback (P := fun (X : TopCat.{0}) ↦
                    TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type)) (X :=
                    LightProfinite.fibre ∞ (π ≫ snd S ℕ∪{∞})) (Y := LightProfinite.fibre ∞ (π ≫ snd S ℕ∪{∞}))
                    (LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞}) ≫ π)
                    (LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞}) ≫ π)).toTop
                (P := fun (X : TopCat.{0}) ↦
                TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type))) :
            Type))))
    (CategoryTheory.Functor.map (lightProfiniteToLightCondSet ⋙ free R) (cover π))
g : obj (lightProfiniteToLightCondSet ⋙ free R) T ⟶ X
r_inf : T ⟶ LightProfinite.fibre ∞ (π ≫ snd S ℕ∪{∞})
σ : S ⟶ LightProfinite.fibre ∞ (π ≫ snd S ℕ∪{∞})
hr : LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞}) ≫ r_inf = 𝟙 (LightProfinite.fibre ∞ (π ≫ snd S ℕ∪{∞}))
this :
  cover π =
    (↑(BinaryCofan.IsColimit.desc'
          (coproductIsColimit (P := fun (X : TopCat.{0}) ↦
            TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type)) T
            (CompHausLike.pullback (P := fun (X : TopCat.{0}) ↦
              TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type)) (X :=
              LightProfinite.fibre ∞ (π ≫ snd S ℕ∪{∞})) (Y := LightProfinite.fibre ∞ (π ≫ snd S ℕ∪{∞}))
              (LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞}) ≫ π) (LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞}) ≫ π)))
          (CompHausLike.pullback.lift (P := fun (X : TopCat.{0}) ↦
            TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type)) π π (𝟙 T) (𝟙 T) ⋯)
          (CompHausLike.pullback.lift (P := fun (X : TopCat.{0}) ↦
            TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type)) π π
            (CompHausLike.pullback.fst (P := fun (X : TopCat.{0}) ↦
                TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type))
                (LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞}) ≫ π) (LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞}) ≫ π) ≫
              LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞}))
            (CompHausLike.pullback.snd (P := fun (X : TopCat.{0}) ↦
                TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type))
                (LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞}) ≫ π) (LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞}) ≫ π) ≫
              LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞}))
            ⋯)) :
      (coproductCocone T
            (CompHausLike.pullback (LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞}) ≫ π)
              (LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞}) ≫ π))).pt ⟶
        CompHausLike.pullback π π)
⊢ Eq (α :=
    obj (free R)
        (lightProfiniteToLightCondSet.obj
          (CompHausLike.pullback (P := fun (X : TopCat.{0}) ↦
            TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type)) (X :=
            LightProfinite.fibre ∞ (π ≫ snd S ℕ∪{∞})) (Y := LightProfinite.fibre ∞ (π ≫ snd S ℕ∪{∞}))
            (LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞}) ≫ π) (LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞}) ≫ π))) ⟶
      X)
    (CategoryTheory.Functor.map (free R)
        (lightProfiniteToLightCondSet.map
          (((((CompHausLike.pullback.fst (P := fun (X : TopCat.{0}) ↦
                      TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type))
                      (LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞}) ≫ π)
                      (LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞}) ≫ π) ≫
                    LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞})) ≫
                  π) ≫
                fst S ℕ∪{∞}) ≫
              σ) ≫
            LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞}))) ≫
      g)
    (CategoryTheory.Functor.map (free R)
        (lightProfiniteToLightCondSet.map
          (((((CompHausLike.pullback.snd (P := fun (X : TopCat.{0}) ↦
                      TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type))
                      (LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞}) ≫ π)
                      (LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞}) ≫ π) ≫
                    LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞})) ≫
                  π) ≫
                fst S ℕ∪{∞}) ≫
              σ) ≫
            LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞}))) ≫
      g)

Tactic:
simp [pullback.condition]

Premises used: CompHausLike.pullback.condition