Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X : C
S : ShortComplex C
hS : S.ShortExact
n₀ n₁ : ℕ
h : n₀ + 1 = n₁
this✝ : HasDerivedCategory C := HasDerivedCategory.standard C
this :
  { X₁ := ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₀).obj hS.singleTriangle.obj₃,
      X₂ := ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₁).obj hS.singleTriangle.obj₁,
      X₃ := ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₁).obj hS.singleTriangle.obj₂,
      f := (preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).homologySequenceδ hS.singleTriangle ↑n₀ ↑n₁ ⋯,
      g := ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₁).map hS.singleTriangle.mor₁,
      zero := ⋯ }.Exact
⊢ { X₁ := AddCommGrpCat.of (Ext X S.X₃ n₀), X₂ := AddCommGrpCat.of (Ext X S.X₁ n₁),
      X₃ := AddCommGrpCat.of (Ext X S.X₂ n₁), f := AddCommGrpCat.ofHom (hS.extClass.postcomp X h),
      g := AddCommGrpCat.ofHom ((mk₀ S.f).postcomp X ⋯), zero := ⋯ }.Exact

Tactic:
ShortComplex.ab_exact_iff_function_exact

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X : C
S : ShortComplex C
hS : S.ShortExact
n₀ n₁ : ℕ
h : n₀ + 1 = n₁
this✝ : HasDerivedCategory C := HasDerivedCategory.standard C
this :
  Function.Exact
    ⇑(ConcreteCategory.hom
        { X₁ := ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₀).obj hS.singleTriangle.obj₃,
            X₂ := ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₁).obj hS.singleTriangle.obj₁,
            X₃ := ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₁).obj hS.singleTriangle.obj₂,
            f :=
              (preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).homologySequenceδ hS.singleTriangle ↑n₀ ↑n₁ ⋯,
            g := ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₁).map hS.singleTriangle.mor₁,
            zero := ⋯ }.f)
    ⇑(ConcreteCategory.hom
        { X₁ := ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₀).obj hS.singleTriangle.obj₃,
            X₂ := ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₁).obj hS.singleTriangle.obj₁,
            X₃ := ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₁).obj hS.singleTriangle.obj₂,
            f :=
              (preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).homologySequenceδ hS.singleTriangle ↑n₀ ↑n₁ ⋯,
            g := ((preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).shift ↑n₁).map hS.singleTriangle.mor₁,
            zero := ⋯ }.g)
⊢ Function.Exact
    ⇑(ConcreteCategory.hom
        { X₁ := AddCommGrpCat.of (Ext X S.X₃ n₀), X₂ := AddCommGrpCat.of (Ext X S.X₁ n₁),
            X₃ := AddCommGrpCat.of (Ext X S.X₂ n₁), f := AddCommGrpCat.ofHom (hS.extClass.postcomp X h),
            g := AddCommGrpCat.ofHom ((mk₀ S.f).postcomp X ⋯), zero := ⋯ }.f)
    ⇑(ConcreteCategory.hom
        { X₁ := AddCommGrpCat.of (Ext X S.X₃ n₀), X₂ := AddCommGrpCat.of (Ext X S.X₁ n₁),
            X₃ := AddCommGrpCat.of (Ext X S.X₂ n₁), f := AddCommGrpCat.ofHom (hS.extClass.postcomp X h),
            g := AddCommGrpCat.ofHom ((mk₀ S.f).postcomp X ⋯), zero := ⋯ }.g)