-- Lean 4 proof step from Mathlib.Algebra.Category.ContinuousCohomology.Basic (goal)
Goal:
R : Type u_1
G : Type u_2
inst : CommRing R
inst_1 : Group G
inst_2 : TopologicalSpace R
inst_3 : TopologicalSpace G
inst_4 : IsTopologicalGroup G
X Y : Action (TopModuleCat R) G
f : X ⟶ Y
⊢ Eq (α :=
    TopModuleCat.ker (M := (obj (homogeneousCochains.{u_1, u_2, u_3} R G) X).X 0) (N :=
        (obj (homogeneousCochains.{u_1, u_2, u_3} R G) X).X ((ComplexShape.up ℕ).next 0))
        ((obj (homogeneousCochains.{u_1, u_2, u_3} R G) X).d 0 ((ComplexShape.up ℕ).next 0)) ⟶
      obj (invariants R G) Y)
    (CategoryStruct.comp (Y := HomologicalComplex.homology (obj (homogeneousCochains.{u_1, u_2, u_3} R G) X) 0)
      (Iso.inv
        (LeftHomologyData.ofIsLimitKernelFork
            (HomologicalComplex.sc (obj (homogeneousCochains.{u_1, u_2, u_3} R G) X) 0) ⋯
            (Limits.KernelFork.ofι
              (TopModuleCat.kerι ((obj (homogeneousCochains.{u_1, u_2, u_3} R G) X).d 0 ((ComplexShape.up ℕ).next 0)))
              ⋯)
            (TopModuleCat.isLimitKer
              ((obj (homogeneousCochains.{u_1, u_2, u_3} R G) X).d 0 ((ComplexShape.up ℕ).next 0)))).homologyIso)
      (CategoryStruct.comp (Y := HomologicalComplex.homology (obj (homogeneousCochains.{u_1, u_2, u_3} R G) Y) 0)
        (homologyMap
          (CategoryTheory.Functor.map (HomologicalComplex.shortComplexFunctor (TopModuleCat R) (ComplexShape.up ℕ) 0)
            (CategoryTheory.Functor.map (homogeneousCochains.{u_1, u_2, u_3} R G) f)))
        (Iso.hom
            (LeftHomologyData.ofIsLimitKernelFork
                (HomologicalComplex.sc (obj (homogeneousCochains.{u_1, u_2, u_3} R G) Y) 0) ⋯
                (Limits.KernelFork.ofι
                  (TopModuleCat.kerι
                    ((obj (homogeneousCochains.{u_1, u_2, u_3} R G) Y).d 0 ((ComplexShape.up ℕ).next 0)))
                  ⋯)
                (TopModuleCat.isLimitKer
                  ((obj (homogeneousCochains.{u_1, u_2, u_3} R G) Y).d 0 ((ComplexShape.up ℕ).next 0)))).homologyIso ≫
          Iso.hom
            (TopModuleCat.ofIso
              (kerHomogeneousCochainsZeroEquiv R G Y ((ComplexShape.up ℕ).next 0)
                continuousCohomologyZeroIso._proof_2)))))
    (Iso.hom
        (TopModuleCat.ofIso
          (kerHomogeneousCochainsZeroEquiv R G X ((ComplexShape.up ℕ).next 0) continuousCohomologyZeroIso._proof_2)) ≫
      CategoryTheory.Functor.map (invariants R G) f)
have rw : Eq (α :=
  (LeftHomologyData.ofIsLimitKernelFork (HomologicalComplex.sc (obj (homogeneousCochains.{u_1, u_2, u_3} R G) X) 0) sorry
        (Limits.KernelFork.ofι
          (TopModuleCat.kerι ((obj (homogeneousCochains.{u_1, u_2, u_3} R G) X).d 0 ((ComplexShape.up ℕ).next 0))) sorry)
        (TopModuleCat.isLimitKer
          ((obj (homogeneousCochains.{u_1, u_2, u_3} R G) X).d 0 ((ComplexShape.up ℕ).next 0)))).K ⟶
    obj (invariants R G) Y)
  (cyclesMap'
      (CategoryTheory.Functor.map (HomologicalComplex.shortComplexFunctor (TopModuleCat R) (ComplexShape.up ℕ) 0)
        (CategoryTheory.Functor.map (homogeneousCochains.{u_1, u_2, u_3} R G) f))
      (LeftHomologyData.ofIsLimitKernelFork (HomologicalComplex.sc (obj (homogeneousCochains.{u_1, u_2, u_3} R G) X) 0)
        sorry
        (Limits.KernelFork.ofι
          (TopModuleCat.kerι ((obj (homogeneousCochains.{u_1, u_2, u_3} R G) X).d 0 ((ComplexShape.up ℕ).next 0))) sorry)
        (TopModuleCat.isLimitKer ((obj (homogeneousCochains.{u_1, u_2, u_3} R G) X).d 0 ((ComplexShape.up ℕ).next 0))))
      (LeftHomologyData.ofIsLimitKernelFork (HomologicalComplex.sc (obj (homogeneousCochains.{u_1, u_2, u_3} R G) Y) 0)
        sorry
        (Limits.KernelFork.ofι
          (TopModuleCat.kerι ((obj (homogeneousCochains.{u_1, u_2, u_3} R G) Y).d 0 ((ComplexShape.up ℕ).next 0))) sorry)
        (TopModuleCat.isLimitKer
          ((obj (homogeneousCochains.{u_1, u_2, u_3} R G) Y).d 0 ((ComplexShape.up ℕ).next 0)))) ≫
    (LeftHomologyData.ofIsLimitKernelFork (HomologicalComplex.sc (obj (homogeneousCochains.{u_1, u_2, u_3} R G) Y) 0) sorry
          (Limits.KernelFork.ofι
            (TopModuleCat.kerι ((obj (homogeneousCochains.{u_1, u_2, u_3} R G) Y).d 0 ((ComplexShape.up ℕ).next 0))) sorry)
          (TopModuleCat.isLimitKer
            ((obj (homogeneousCochains.{u_1, u_2, u_3} R G) Y).d 0 ((ComplexShape.up ℕ).next 0)))).π ≫
      Iso.hom
        (TopModuleCat.ofIso
          (kerHomogeneousCochainsZeroEquiv R G Y ((ComplexShape.up ℕ).next 0) continuousCohomologyZeroIso._proof_2)))
  ((LeftHomologyData.ofIsLimitKernelFork (HomologicalComplex.sc (obj (homogeneousCochains.{u_1, u_2, u_3} R G) X) 0) sorry
        (Limits.KernelFork.ofι
          (TopModuleCat.kerι ((obj (homogeneousCochains.{u_1, u_2, u_3} R G) X).d 0 ((ComplexShape.up ℕ).next 0))) sorry)
        (TopModuleCat.isLimitKer
          ((obj (homogeneousCochains.{u_1, u_2, u_3} R G) X).d 0 ((ComplexShape.up ℕ).next 0)))).π ≫
    Iso.hom
        (TopModuleCat.ofIso
          (kerHomogeneousCochainsZeroEquiv R G X ((ComplexShape.up ℕ).next 0) continuousCohomologyZeroIso._proof_2)) ≫
      CategoryTheory.Functor.map (invariants R G) f)
Tactic: rw [LeftHomologyData.leftHomologyIso_inv_naturality_assoc, Iso.inv_hom_id_assoc,
    ← cancel_epi (LeftHomologyData.π _), leftHomologyπ_naturality'_assoc]