-- 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∪{∞}))
refine :
  Eq (α :=
    obj
        (parallelPair (X :=
            lightProfiniteToLightCondSet.obj
              (CompHausLike.pullback (P := fun (X : TopCat.{0}) ↦
                TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type)) π π))
            (lightProfiniteToLightCondSet.map
              (CompHausLike.pullback.fst (P := fun (X : TopCat.{0}) ↦
                TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type)) π π))
            (lightProfiniteToLightCondSet.map
              (CompHausLike.pullback.snd (P := fun (X : TopCat.{0}) ↦
                TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type)) π π)) ⋙
          free R)
        WalkingParallelPair.zero ⟶
      X)
    (CategoryTheory.Functor.map
        (parallelPair (X :=
            lightProfiniteToLightCondSet.obj
              (CompHausLike.pullback (P := fun (X : TopCat.{0}) ↦
                TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type)) π π))
            (lightProfiniteToLightCondSet.map
              (CompHausLike.pullback.fst (P := fun (X : TopCat.{0}) ↦
                TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type)) π π))
            (lightProfiniteToLightCondSet.map
              (CompHausLike.pullback.snd (P := fun (X : TopCat.{0}) ↦
                TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type)) π π)) ⋙
          free R)
        WalkingParallelPairHom.left ≫
      (g -
          CategoryTheory.Functor.map (lightProfiniteToLightCondSet ⋙ free R)
              (r_inf ≫ LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞})) ≫
            g +
        CategoryTheory.Functor.map (lightProfiniteToLightCondSet ⋙ free R)
            (r_inf ≫
              LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞}) ≫
                π ≫ fst S ℕ∪{∞} ≫ σ ≫ LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞})) ≫
          g))
    (CategoryTheory.Functor.map
        (parallelPair (X :=
            lightProfiniteToLightCondSet.obj
              (CompHausLike.pullback (P := fun (X : TopCat.{0}) ↦
                TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type)) π π))
            (lightProfiniteToLightCondSet.map
              (CompHausLike.pullback.fst (P := fun (X : TopCat.{0}) ↦
                TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type)) π π))
            (lightProfiniteToLightCondSet.map
              (CompHausLike.pullback.snd (P := fun (X : TopCat.{0}) ↦
                TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type)) π π)) ⋙
          free R)
        WalkingParallelPairHom.right ≫
      (g -
          CategoryTheory.Functor.map (lightProfiniteToLightCondSet ⋙ free R)
              (r_inf ≫ LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞})) ≫
            g +
        CategoryTheory.Functor.map (lightProfiniteToLightCondSet ⋙ free R)
            (r_inf ≫
              LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞}) ≫
                π ≫ fst S ℕ∪{∞} ≫ σ ≫ LightProfinite.fibreIncl ∞ (π ≫ snd S ℕ∪{∞})) ≫
          g))
⊢ Cocone (C := LightCondMod R)
    (parallelPair (X :=
        lightProfiniteToLightCondSet.obj
          (CompHausLike.pullback (P := fun (X : TopCat.{0}) ↦
            TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type)) π π))
        (lightProfiniteToLightCondSet.map
          (CompHausLike.pullback.fst (P := fun (X : TopCat.{0}) ↦
            TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type)) π π))
        (lightProfiniteToLightCondSet.map
          (CompHausLike.pullback.snd (P := fun (X : TopCat.{0}) ↦
            TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type)) π π)) ⋙
      free R)

Tactic:
refine Cocone.ofCofork (Cofork.ofπ (g -
    (lightProfiniteToLightCondSet ⋙ free R).map
      (r_inf ≫ LightProfinite.fibreIncl ∞ (π ≫ snd _ _)) ≫ g +
    (lightProfiniteToLightCondSet ⋙ free R).map
      (r_inf ≫ LightProfinite.fibreIncl ∞ (π ≫ snd _ _) ≫ π ≫ fst _ _ ≫ σ ≫
        LightProfinite.fibreIncl ∞ (π ≫ snd _ _)) ≫ g) ?_)

Premises used: CategoryTheory.Limits.Cofork.ofπ, CategoryTheory.Limits.Cocone.ofCofork