-- Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.Map
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 : HasExt.{w, v, u} C
inst_8 : HasExt.{w', v', u'} D
X Y : C
f : X ⟶ Y
rw :
  Eq.{w' + 1} (α := Ext.{w', v', u'} (F.obj X) (F.obj Y) 0)
    (SmallShiftedHom.mk₀ (HomologicalComplex.quasiIso D (ComplexShape.up ℤ)) 0 ⋯
      (CategoryStruct.comp (Y :=
        Functor.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)))
          (Functor.obj (CochainComplex.singleFunctor C 0) X))
        (Iso.inv (Iso.app (F.mapCochainComplexSingleFunctor 0) X) :
          Quiver.Hom (V := HomologicalComplex.{v', u', 0} D (ComplexShape.up ℤ))
            (Functor.obj (F ⋙ CochainComplex.singleFunctor D 0) X)
            (Functor.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)))
              (Functor.obj (CochainComplex.singleFunctor C 0) X)))
        (Functor.map
              (LocalizerMorphism.functor (W₁ := HomologicalComplex.quasiIso C (ComplexShape.up ℤ)) (W₂ :=
                HomologicalComplex.quasiIso D (ComplexShape.up ℤ))
                (F.mapHomologicalComplexUpToQuasiIsoLocalizerMorphism (ComplexShape.up ℤ)))
              (Functor.map (CochainComplex.singleFunctor C 0) f) ≫
            Iso.hom (Iso.app (F.mapCochainComplexSingleFunctor 0) Y) :
          Quiver.Hom (V := HomologicalComplex.{v', u', 0} D (ComplexShape.up ℤ))
            (Functor.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)))
              (Functor.obj (CochainComplex.singleFunctor C 0) X))
            (Functor.obj (F ⋙ CochainComplex.singleFunctor D 0) Y))))
    (SmallShiftedHom.mk₀ (HomologicalComplex.quasiIso D (ComplexShape.up ℤ)) 0 mk₀._proof_1
      (Functor.map (CochainComplex.singleFunctor D 0) (F.map f)))
⊢ Eq.{w' + 1} (α := Ext.{w', v', u'} (F.obj X) (F.obj Y) 0)
    (LocalizerMorphism.smallShiftedHomMap (W₁ := HomologicalComplex.quasiIso C (ComplexShape.up ℤ)) (W₂ :=
      HomologicalComplex.quasiIso D (ComplexShape.up ℤ))
      (F.mapHomologicalComplexUpToQuasiIsoLocalizerMorphism (ComplexShape.up ℤ)) (X₁ :=
      Functor.obj (CochainComplex.singleFunctor C 0) X) (Iso.app (F.mapCochainComplexSingleFunctor 0) X)
      (Iso.app (F.mapCochainComplexSingleFunctor 0) Y)
      (SmallShiftedHom.mk₀ (HomologicalComplex.quasiIso C (ComplexShape.up ℤ)) 0 mk₀._proof_1
        (Functor.map (CochainComplex.singleFunctor C 0) f)))
    (SmallShiftedHom.mk₀ (HomologicalComplex.quasiIso D (ComplexShape.up ℤ)) 0 mk₀._proof_1
      (Functor.map (CochainComplex.singleFunctor D 0) (F.map f)))

Tactic:
rw [(F.mapHomologicalComplexUpToQuasiIsoLocalizerMorphism (.up ℤ)).smallShiftedHomMap_mk₀
    ((F.mapCochainComplexSingleFunctor 0).app X) ((F.mapCochainComplexSingleFunctor 0).app Y)
    (0 : ℤ) rfl]

Premises used: CategoryTheory.Functor.mapHomologicalComplexUpToQuasiIsoLocalizerMorphism, CategoryTheory.Functor.mapCochainComplexSingleFunctor, Int, CategoryTheory.Iso.app, OfNat.ofNat, CategoryTheory.LocalizerMorphism.smallShiftedHomMap_mk₀, rfl, ComplexShape.up