∀ {k G H : Type u} [inst : CommRing k] [inst_1 : Group G] [inst_2 : Group H] {A : Rep k H} {B : Rep k G} (f : G →* H)
  (φ : (Action.res (ModuleCat k) f).obj A ⟶ B) (x : ↥(groupCohomology.cocycles₁ A)),
  (ModuleCat.Hom.hom (groupCohomology.shortComplexH1 B).g).ker.subtype
      ((CategoryTheory.ConcreteCategory.hom (groupCohomology.mapCocycles₁ f φ)) x) =
    ((ModuleCat.Hom.hom φ.hom).compLeft G)
      ((LinearMap.funLeft k ↑A.V ⇑f) ((ModuleCat.Hom.hom (groupCohomology.shortComplexH1 A).g).ker.subtype x))