-- Module: Mathlib.Condensed.Light.Sequence
Goal:
R : Type
inst : CommRing R
X Y : LightCondMod R
p : X ⟶ Y
hp : Epi p
S : LightProfinite
f :
  obj (free R)
      (tensorObj (𝒞 :=
          CompHausLike.category fun (X : TopCat.{0}) ↦
            TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type))
          S ℕ∪{∞}).toCondensed ⟶
    Y
T : LightProfinite
π :
  T ⟶
    tensorObj (𝒞 :=
      CompHausLike.category fun (X : TopCat.{0}) ↦
        TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type))
      S ℕ∪{∞}
g : obj (free R) T.toCondensed ⟶ X
hπ : Epi π
comm : CategoryTheory.Functor.map (lightProfiniteToLightCondSet ⋙ free R) π ≫ f = g ≫ p
S' T' : LightProfinite
y' : S' ⟶ S
π' :
  T' ⟶
    tensorObj (𝒞 :=
      CompHausLike.category fun (X : TopCat.{0}) ↦
        TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type))
      S' ℕ∪{∞}
g' : T' ⟶ T
hπ'_1 : Epi π'
hy' : Epi y'
comp : π' ≫ y' ▷ ℕ∪{∞} = g' ≫ π
epi : Epi (cover π')
split :
  SplitEpi (X := LightProfinite.fibre ∞ (π' ≫ snd S' ℕ∪{∞}))
    (LightProfinite.fibreIncl ∞ (π' ≫ snd S' ℕ∪{∞}) ≫ π' ≫ fst S' ℕ∪{∞})
hS' :
  Nonempty
    (↑(S'.toTop (P := fun (X : TopCat.{0}) ↦
          TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type))) :
      Type)
this_1 : Mono (X := LightProfinite.fibre ∞ (π' ≫ snd S' ℕ∪{∞})) (LightProfinite.fibreIncl ∞ (π' ≫ snd S' ℕ∪{∞}))
this_2 :
  Nonempty
    (↑((LightProfinite.fibre ∞ (π' ≫ snd S' ℕ∪{∞})).toTop (P := fun (X : TopCat.{0}) ↦
          TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type))) :
      Type)
r_inf : T' ⟶ LightProfinite.fibre ∞ (π' ≫ snd S' ℕ∪{∞})
hr : LightProfinite.fibreIncl ∞ (π' ≫ snd S' ℕ∪{∞}) ≫ r_inf = 𝟙 (LightProfinite.fibre ∞ (π' ≫ snd S' ℕ∪{∞}))
hπ'' : RegularEpi (lightProfiniteToLightCondSet.map π') := regularEpiOfEpi (lightProfiniteToLightCondSet.map π')
this : EffectiveEpi π'
hπ' : RegularEpi (lightProfiniteToLightCondSet.map π') :=
  regularEpiOfPreserves π' lightProfiniteToLightCondSet
    (pullback.cone (P := fun (X : TopCat.{0}) ↦
      TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type)) π' π')
    (pullback.isLimit (P := fun (X : TopCat.{0}) ↦
      TotallyDisconnectedSpace (↑X : Type) ∧ SecondCountableTopology (↑X : Type)) π' π')
hc :
  Limits.IsColimit (C := LightCondMod R) (F := Limits.parallelPair hπ'.left hπ'.right ⋙ free R)
    (mapCocone (free R) (Limits.Cofork.ofπ (lightProfiniteToLightCondSet.map π') ⋯))
pos :
  Eq (α := obj (Limits.parallelPair hπ'.left hπ'.right ⋙ free R) Limits.WalkingParallelPair.one ⟶ Y)
    (CategoryTheory.Functor.map (lightProfiniteToLightCondSet ⋙ free R) (π' ≫ y' ▷ ℕ∪{∞}) ≫ f :
      obj (lightProfiniteToLightCondSet ⋙ free R) T' ⟶ Y)
    (CategoryStruct.comp (Y :=
        obj
          (obj (Functor.const Limits.WalkingParallelPair)
            (Limits.Cocone.pt (F := Limits.parallelPair hπ'.left hπ'.right ⋙ free R)
              (mapCocone (free R) (Limits.Cofork.ofπ (lightProfiniteToLightCondSet.map π') ⋯))))
          Limits.WalkingParallelPair.one)
        (NatTrans.app (Limits.Cocone.ι (mapCocone (free R) (Limits.Cofork.ofπ (lightProfiniteToLightCondSet.map π') ⋯)))
          Limits.WalkingParallelPair.one)
        (hc.desc
          (cocone R (X := X) π' (CategoryTheory.Functor.map (lightProfiniteToLightCondSet ⋙ free R) g' ≫ g) r_inf
            split.section_ hr)) ≫
      p)
⊢ Eq (α := obj (lightProfiniteToLightCondSet ⋙ free R) T' ⟶ Y)
    (CategoryTheory.Functor.map (lightProfiniteToLightCondSet ⋙ free R) (π' ≫ y' ▷ ℕ∪{∞}) ≫ f)
    (CategoryTheory.Functor.map (lightProfiniteToLightCondSet ⋙ free R) π' ≫
      hc.desc
          (cocone R (X := X) π' (CategoryTheory.Functor.map (lightProfiniteToLightCondSet ⋙ free R) g' ≫ g) r_inf
            split.section_ hr) ≫
        p)

Tactic:
change _ = (((free R).mapCocone _).ι.app .one ≫
      hc.desc (cocone R π' _ r_inf split.section_ hr)) ≫ p

Premises used: CategoryTheory.Functor.mapCocone, CategoryTheory.SplitEpi.section_, CategoryTheory.Limits.Cocone.ι, _private.0.InternalProjectivityProof.cocone, CategoryTheory.Limits.IsColimit.desc, CategoryTheory.CategoryStruct.comp, LightCondensed.free, CategoryTheory.NatTrans.app