-- 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 :
  CategoryTheory.Functor.map (lightProfiniteToLightCondSet ⋙ free R) (π' ≫ y' ▷ ℕ∪{∞}) ≫ f =
    HAdd.hAdd (β := obj (free R) (lightProfiniteToLightCondSet.obj T') ⟶ Y)
      (HSub.hSub (α := obj (free R) (lightProfiniteToLightCondSet.obj T') ⟶ Y) (β :=
        obj (free R) (lightProfiniteToLightCondSet.obj T') ⟶ Y)
        (CategoryTheory.Functor.map (lightProfiniteToLightCondSet ⋙ free R) (π' ≫ y' ▷ ℕ∪{∞}) ≫ f :
          obj (free R) (lightProfiniteToLightCondSet.obj T') ⟶ Y)
        (CategoryTheory.Functor.map (lightProfiniteToLightCondSet ⋙ free R)
              (((r_inf ≫ LightProfinite.fibreIncl ∞ (π' ≫ snd S' ℕ∪{∞})) ≫ π') ≫ y' ▷ ℕ∪{∞}) ≫
            f :
          obj (free R) (lightProfiniteToLightCondSet.obj T') ⟶ Y))
      (CategoryTheory.Functor.map (lightProfiniteToLightCondSet ⋙ free R)
            (((((((r_inf ≫ LightProfinite.fibreIncl ∞ (π' ≫ snd S' ℕ∪{∞})) ≫ π') ≫ fst S' ℕ∪{∞}) ≫ split.section_) ≫
                  LightProfinite.fibreIncl ∞ (π' ≫ snd S' ℕ∪{∞})) ≫
                π') ≫
              y' ▷ ℕ∪{∞}) ≫
          f :
        obj (free R) (lightProfiniteToLightCondSet.obj T') ⟶ Y)
⊢ CategoryTheory.Functor.map (free R) (lightProfiniteToLightCondSet.map π') ≫
      CategoryTheory.Functor.map (free R) (lightProfiniteToLightCondSet.map (y' ▷ ℕ∪{∞})) ≫ f =
    CategoryTheory.Functor.map (free R) (lightProfiniteToLightCondSet.map g') ≫
          CategoryTheory.Functor.map (free R) (lightProfiniteToLightCondSet.map π) ≫ f -
        CategoryTheory.Functor.map (free R) (lightProfiniteToLightCondSet.map r_inf) ≫
          CategoryTheory.Functor.map (free R)
              (lightProfiniteToLightCondSet.map (LightProfinite.fibreIncl ∞ (π' ≫ snd S' ℕ∪{∞}))) ≫
            CategoryTheory.Functor.map (free R) (lightProfiniteToLightCondSet.map g') ≫
              CategoryTheory.Functor.map (free R) (lightProfiniteToLightCondSet.map π) ≫ f +
      CategoryTheory.Functor.map (free R) (lightProfiniteToLightCondSet.map r_inf) ≫
        CategoryTheory.Functor.map (free R)
            (lightProfiniteToLightCondSet.map (LightProfinite.fibreIncl ∞ (π' ≫ snd S' ℕ∪{∞}))) ≫
          CategoryTheory.Functor.map (free R) (lightProfiniteToLightCondSet.map π') ≫
            CategoryTheory.Functor.map (free R) (lightProfiniteToLightCondSet.map (fst S' ℕ∪{∞})) ≫
              CategoryTheory.Functor.map (free R) (lightProfiniteToLightCondSet.map split.section_) ≫
                CategoryTheory.Functor.map (free R)
                    (lightProfiniteToLightCondSet.map (LightProfinite.fibreIncl ∞ (π' ≫ snd S' ℕ∪{∞}))) ≫
                  CategoryTheory.Functor.map (free R) (lightProfiniteToLightCondSet.map g') ≫
                    CategoryTheory.Functor.map (free R) (lightProfiniteToLightCondSet.map π) ≫ f

Tactic:
simp only [← Functor.map_comp, ← Functor.comp_map, ← assoc, ← comp]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Functor.map_comp, CategoryTheory.Functor.comp_map