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
S : ShortComplex C
hS : S.ShortExact
Y : C
n : ℕ
this✝ : HasDerivedCategory C := HasDerivedCategory.standard C
this :
  {
      X₁ :=
        ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n).obj
          ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₁,
      X₂ :=
        ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n).obj
          ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₂,
      X₃ :=
        ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n).obj
          ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₃,
      f :=
        ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n).map
          ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).mor₁,
      g :=
        ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n).map
          ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).mor₂,
      zero := ⋯ }.Exact
⊢ { X₁ := AddCommGrpCat.of (Ext S.X₃ Y n), X₂ := AddCommGrpCat.of (Ext S.X₂ Y n), X₃ := AddCommGrpCat.of (Ext S.X₁ Y n),
      f := AddCommGrpCat.ofHom ((mk₀ S.g).precomp Y ⋯), g := AddCommGrpCat.ofHom ((mk₀ S.f).precomp Y ⋯),
      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
S : ShortComplex C
hS : S.ShortExact
Y : C
n : ℕ
this✝ : HasDerivedCategory C := HasDerivedCategory.standard C
this :
  Function.Exact
    ⇑(ConcreteCategory.hom
        {
            X₁ :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n).obj
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₁,
            X₂ :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n).obj
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₂,
            X₃ :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n).obj
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₃,
            f :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n).map
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).mor₁,
            g :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n).map
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).mor₂,
            zero := ⋯ }.f)
    ⇑(ConcreteCategory.hom
        {
            X₁ :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n).obj
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₁,
            X₂ :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n).obj
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₂,
            X₃ :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n).obj
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₃,
            f :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n).map
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).mor₁,
            g :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n).map
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).mor₂,
            zero := ⋯ }.g)
⊢ Function.Exact
    ⇑(ConcreteCategory.hom
        { X₁ := AddCommGrpCat.of (Ext S.X₃ Y n), X₂ := AddCommGrpCat.of (Ext S.X₂ Y n),
            X₃ := AddCommGrpCat.of (Ext S.X₁ Y n), f := AddCommGrpCat.ofHom ((mk₀ S.g).precomp Y ⋯),
            g := AddCommGrpCat.ofHom ((mk₀ S.f).precomp Y ⋯), zero := ⋯ }.f)
    ⇑(ConcreteCategory.hom
        { X₁ := AddCommGrpCat.of (Ext S.X₃ Y n), X₂ := AddCommGrpCat.of (Ext S.X₂ Y n),
            X₃ := AddCommGrpCat.of (Ext S.X₁ Y n), f := AddCommGrpCat.ofHom ((mk₀ S.g).precomp Y ⋯),
            g := AddCommGrpCat.ofHom ((mk₀ S.f).precomp Y ⋯), zero := ⋯ }.g)