-- Module: Mathlib.CategoryTheory.Abelian.Projective.Ext
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
X Y : C
R : ProjectiveResolution.{v, u} X
n : ℕ
inst_3 : HasDerivedCategory.{u_1, v, u} C
x : Cocycle R.cochainComplex (Functor.obj (singleFunctor C 0) Y) (↑n : ℤ)
change :
  Eq.{u_1 + 1} (α :=
    ShiftedHom (C := DerivedCategory.{u_1, v, u} C) (M := ℤ)
      (Functor.obj DerivedCategory.Q (Functor.obj (singleFunctor C 0) X))
      (Functor.obj DerivedCategory.Q (Functor.obj (singleFunctor C 0) Y)) (↑n : ℤ))
    ((SmallShiftedHom.equiv (HomologicalComplex.quasiIso C (ComplexShape.up ℤ)) DerivedCategory.Q :
          (a :
              SmallShiftedHom (HomologicalComplex.quasiIso C (ComplexShape.up ℤ)) ((singleFunctor C 0).obj X)
                ((singleFunctor C 0).obj Y) (↑n : ℤ)) →
            ShiftedHom (DerivedCategory.Q.obj ((singleFunctor C 0).obj X))
              (DerivedCategory.Q.obj ((singleFunctor C 0).obj Y)) (↑n : ℤ))
        (SmallShiftedHom.comp (W := HomologicalComplex.quasiIso C (ComplexShape.up ℤ))
          (SmallShiftedHom.mk₀Inv (W := HomologicalComplex.quasiIso C (ComplexShape.up ℤ)) 0 ⋯ R.π' ⋯)
          (CohomologyClass.mk x).toSmallShiftedHom ⋯) :
      ShiftedHom (C := DerivedCategory.{u_1, v, u} C) (M := ℤ)
        (Functor.obj DerivedCategory.Q (Functor.obj (singleFunctor C 0) X))
        (Functor.obj DerivedCategory.Q (Functor.obj (singleFunctor C 0) Y)) (↑n : ℤ))
    (ShiftedHom.comp
        (ShiftedHom.mk₀ 0 ⋯
          (NatTrans.app (Iso.hom (DerivedCategory.singleFunctorIsoCompQ C 0)) X ≫
            inv (Functor.map DerivedCategory.Q R.π')))
        (ShiftedHom.comp
          (ShiftedHom.map
            ((AddEquiv.symm Cocycle.equivHomShift :
                (a : Cocycle R.cochainComplex ((singleFunctor C 0).obj Y) (↑n : ℤ)) →
                  R.cochainComplex ⟶ (shiftFunctor (CochainComplex C ℤ) (↑n : ℤ)).obj ((singleFunctor C 0).obj Y))
              x)
            DerivedCategory.Q)
          (ShiftedHom.mk₀ 0 ⋯ (NatTrans.app (Iso.inv (DerivedCategory.singleFunctorIsoCompQ C 0)) Y)) ⋯)
        ⋯ :
      ShiftedHom (C := DerivedCategory.{u_1, v, u} C) (M := ℤ) (Functor.obj (DerivedCategory.singleFunctor C 0) X)
        (Functor.obj (DerivedCategory.singleFunctor C 0) Y) (↑n : ℤ))
⊢ Eq.{u_1 + 1} (α :=
    ShiftedHom (C := DerivedCategory.{u_1, v, u} C) (M := ℤ) (Functor.obj (DerivedCategory.singleFunctor C 0) X)
      (Functor.obj (DerivedCategory.singleFunctor C 0) Y) (↑n : ℤ))
    (Ext.hom.{u_1, w, v, u}
      ((Equiv.symm R.extEquivCohomologyClass :
          (a : CohomologyClass R.cochainComplex ((singleFunctor C 0).obj Y) (↑n : ℤ)) → Ext X Y n)
        (CohomologyClass.mk x)))
    (ShiftedHom.comp
      (ShiftedHom.mk₀ 0 ⋯
        (NatTrans.app (Iso.hom (DerivedCategory.singleFunctorIsoCompQ C 0)) X ≫
          inv (Functor.map DerivedCategory.Q R.π')))
      (ShiftedHom.comp
        (ShiftedHom.map
          ((AddEquiv.symm Cocycle.equivHomShift :
              (a : Cocycle R.cochainComplex ((singleFunctor C 0).obj Y) (↑n : ℤ)) →
                R.cochainComplex ⟶ (shiftFunctor (CochainComplex C ℤ) (↑n : ℤ)).obj ((singleFunctor C 0).obj Y))
            x)
          DerivedCategory.Q)
        (ShiftedHom.mk₀ 0 ⋯ (NatTrans.app (Iso.inv (DerivedCategory.singleFunctorIsoCompQ C 0)) Y)) ⋯)
      ⋯)

Tactic:
change SmallShiftedHom.equiv _ _ (.comp _ (CohomologyClass.mk x).toSmallShiftedHom _) = _

Premises used: CochainComplex.HomComplex.CohomologyClass.mk, CochainComplex.HomComplex.CohomologyClass.toSmallShiftedHom, CategoryTheory.Localization.SmallShiftedHom.comp, CategoryTheory.Localization.SmallShiftedHom.equiv, DFunLike.coe