-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Map (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
D : Type u'
inst_2 : Category.{v', u'} D
inst_3 : Abelian.{v', u'} D
F : Functor.{v, v', u, u'} C D
inst_4 : F.Additive
inst_5 : PreservesFiniteLimits F
inst_6 : PreservesFiniteColimits F
inst_7 : HasDerivedCategory.{t, v, u} C
inst_8 : HasDerivedCategory.{t', v', u'} D
inst_9 : HasExt.{w, v, u} C
inst_10 : HasExt.{w', v', u'} D
X Y : C
n : ℕ
e : Ext.{w, v, u} X Y n
⊢ Eq.{t' + 1} (α :=
    ShiftedHom (C := DerivedCategory.{t', v', u'} D) (M := ℤ) (obj (DerivedCategory.singleFunctor D 0) (F.obj X))
      (obj (DerivedCategory.singleFunctor D 0) (F.obj Y)) (↑n : ℤ))
    (hom.{t', w', v', u'} (mapExactFunctor F e))
    (CategoryStruct.comp (Y :=
      obj (Functor.comp (D := DerivedCategory.{t, v, u} C) (DerivedCategory.singleFunctor C 0) F.mapDerivedCategory) X)
      (NatTrans.app (Iso.inv (F.mapDerivedCategorySingleFunctor 0)) X)
      (ShiftedHom.map e.hom F.mapDerivedCategory ≫
        map (shiftFunctor (DerivedCategory.{t', v', u'} D) (A := ℤ) (↑n : ℤ))
          (NatTrans.app (Iso.hom (F.mapDerivedCategorySingleFunctor 0)) Y)))
have this : Eq.{t' + 1} (α :=
  ShiftedHom (C := DerivedCategory.{t', v', u'} D) (M := ℤ) (obj (DerivedCategory.singleFunctor D 0) (F.obj X))
    (obj (DerivedCategory.singleFunctor D 0) (F.obj Y)) (↑n : ℤ))
  (hom.{t', w', v', u'} (mapExactFunctor F e))
  (ShiftedHom.comp (Y :=
    obj (C := HomologicalComplex.{v, u, 0} C (ComplexShape.up ℤ))
      (Functor.comp (D := DerivedCategory.{t, v, u} C) DerivedCategory.Q F.mapDerivedCategory)
      (obj (CochainComplex.singleFunctor C 0) X))
    (ShiftedHom.mk₀ 0 sorry
      (CategoryStruct.comp (Y :=
        obj DerivedCategory.Q
          (obj (C := HomologicalComplex.{v, u, 0} C (ComplexShape.up ℤ))
            (LocalizerMorphism.functor (W₁ := HomologicalComplex.quasiIso C (ComplexShape.up ℤ)) (W₂ :=
              HomologicalComplex.quasiIso D (ComplexShape.up ℤ))
              (F.mapHomologicalComplexUpToQuasiIsoLocalizerMorphism (ComplexShape.up ℤ)))
            (obj (CochainComplex.singleFunctor C 0) X)))
        (map DerivedCategory.Q (Iso.inv (Iso.app (F.mapCochainComplexSingleFunctor 0) X)) :
          Quiver.Hom (V := DerivedCategory.{t', v', u'} D)
            (obj DerivedCategory.Q (obj (F ⋙ CochainComplex.singleFunctor D 0) X))
            (obj DerivedCategory.Q
              (obj (C := HomologicalComplex.{v, u, 0} C (ComplexShape.up ℤ))
                (LocalizerMorphism.functor (W₁ := HomologicalComplex.quasiIso C (ComplexShape.up ℤ)) (W₂ :=
                  HomologicalComplex.quasiIso D (ComplexShape.up ℤ))
                  (F.mapHomologicalComplexUpToQuasiIsoLocalizerMorphism (ComplexShape.up ℤ)))
                (obj (CochainComplex.singleFunctor C 0) X))))
        (NatTrans.app (Iso.hom (Iso.symm F.mapDerivedCategoryFactors)) (obj (CochainComplex.singleFunctor C 0) X) :
          Quiver.Hom (V := DerivedCategory.{t', v', u'} D)
            (obj (C := HomologicalComplex.{v, u, 0} C (ComplexShape.up ℤ))
              (LocalizerMorphism.functor (W₁ := HomologicalComplex.quasiIso C (ComplexShape.up ℤ)) (W₂ :=
                  HomologicalComplex.quasiIso D (ComplexShape.up ℤ))
                  (F.mapHomologicalComplexUpToQuasiIsoLocalizerMorphism (ComplexShape.up ℤ)) ⋙
                DerivedCategory.Q)
              (obj (CochainComplex.singleFunctor C 0) X))
            (obj (C := HomologicalComplex.{v, u, 0} C (ComplexShape.up ℤ))
              (Functor.comp (D := DerivedCategory.{t, v, u} C) DerivedCategory.Q F.mapDerivedCategory)
              (obj (CochainComplex.singleFunctor C 0) X)))))
    (ShiftedHom.comp (Y :=
      obj (C := DerivedCategory.{t, v, u} C) F.mapDerivedCategory
        (obj DerivedCategory.Q (obj (CochainComplex.singleFunctor C 0) Y)))
      (ShiftedHom.map
        ((SmallShiftedHom.equiv (HomologicalComplex.quasiIso C (ComplexShape.up ℤ)) DerivedCategory.Q :
            (a :
                SmallShiftedHom (HomologicalComplex.quasiIso C (ComplexShape.up ℤ))
                  ((CochainComplex.singleFunctor C 0).obj X) ((CochainComplex.singleFunctor C 0).obj Y) (↑n : ℤ)) →
              ShiftedHom (DerivedCategory.Q.obj ((CochainComplex.singleFunctor C 0).obj X))
                (DerivedCategory.Q.obj ((CochainComplex.singleFunctor C 0).obj Y)) (↑n : ℤ))
          e)
        F.mapDerivedCategory)
      (ShiftedHom.mk₀ 0 sorry
        (NatTrans.app (Iso.inv (Iso.symm F.mapDerivedCategoryFactors)) (obj (CochainComplex.singleFunctor C 0) Y) ≫
          map DerivedCategory.Q (Iso.hom (Iso.app (F.mapCochainComplexSingleFunctor 0) Y))))
      sorry)
    sorry)
Tactic: have : (e.mapExactFunctor F).hom = _ :=
    ((F.mapHomologicalComplexUpToQuasiIsoLocalizerMorphism
      (ComplexShape.up ℤ)).equiv_smallShiftedHomMap DerivedCategory.Q DerivedCategory.Q
        ((F.mapCochainComplexSingleFunctor 0).app X) ((F.mapCochainComplexSingleFunctor 0).app Y)
          F.mapDerivedCategory F.mapDerivedCategoryFactors.symm e)