-- 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 := HomologicalComplex.homology (obj (homogeneousCochains.{u_1, u_2, u_3} R G) X) 0 ⟶ obj (invariants R G) Y)
    (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
          (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 ≫
        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 :=
  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)
          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)))).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) 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)))).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)
Tactic: rw [Category.assoc, ← Iso.inv_comp_eq]