-- Lean 4 proof step from Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasBinaryBiproducts.{v_1, u_1} C
inst_4 : Preadditive.{v_2, u_2} D
inst_5 : HasBinaryBiproducts.{v_2, u_2} D
K L : CochainComplex.{v_1, u_1, 0} C ℤ
φ : K ⟶ L
G : CategoryTheory.Functor.{v_1, v_2, u_1, u_2} C D
inst_6 : G.Additive
⊢ Eq (α :=
    Quiver.Hom (V := HomologicalComplex.{v_2, u_2, 0} D (ComplexShape.up ℤ))
      (Triangle.obj₃ (Functor.obj (Functor.mapTriangle (G.mapHomologicalComplex (ComplexShape.up ℤ))) (triangle φ)))
      (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v_2, u_2, 0} D (ComplexShape.up ℤ)) (A := ℤ) 1)
        (Triangle.obj₁
          (triangle (K := Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) K)
            (CategoryTheory.Functor.map (G.mapHomologicalComplex (ComplexShape.up ℤ)) φ)))))
    (Triangle.mor₃ (Functor.obj (Functor.mapTriangle (G.mapHomologicalComplex (ComplexShape.up ℤ))) (triangle φ)) ≫
      CategoryTheory.Functor.map
        (CategoryTheory.shiftFunctor (HomologicalComplex.{v_2, u_2, 0} D (ComplexShape.up ℤ)) (A := ℤ) 1)
        (Iso.hom
          (Iso.refl
            (Triangle.obj₁
              (Functor.obj (Functor.mapTriangle (G.mapHomologicalComplex (ComplexShape.up ℤ))) (triangle φ))))))
    (CategoryStruct.comp (Y :=
      Triangle.obj₃
        (triangle (K := Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) K)
          (CategoryTheory.Functor.map (G.mapHomologicalComplex (ComplexShape.up ℤ)) φ)))
      (Iso.hom (mapHomologicalComplexIso φ G) :
        Quiver.Hom (V := HomologicalComplex.{v_2, u_2, 0} D (ComplexShape.up ℤ))
          (Triangle.obj₃ (Functor.obj (Functor.mapTriangle (G.mapHomologicalComplex (ComplexShape.up ℤ))) (triangle φ)))
          (Triangle.obj₃
            (triangle (K := Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) K)
              (CategoryTheory.Functor.map (G.mapHomologicalComplex (ComplexShape.up ℤ)) φ))))
      (Triangle.mor₃
          (triangle (K := Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) K)
            (CategoryTheory.Functor.map (G.mapHomologicalComplex (ComplexShape.up ℤ)) φ)) :
        Quiver.Hom (V := HomologicalComplex.{v_2, u_2, 0} D (ComplexShape.up ℤ))
          (Triangle.obj₃
            (triangle (K := Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) K)
              (CategoryTheory.Functor.map (G.mapHomologicalComplex (ComplexShape.up ℤ)) φ)))
          (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v_2, u_2, 0} D (ComplexShape.up ℤ)) (A := ℤ) 1)
            (Triangle.obj₁
              (triangle (K := Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) K)
                (CategoryTheory.Functor.map (G.mapHomologicalComplex (ComplexShape.up ℤ)) φ))))))
have refine_3 : Eq (α :=
  Quiver.Hom (V := HomologicalComplex.{v_2, u_2, 0} D (ComplexShape.up ℤ))
    (Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) (mappingCone φ))
    (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v_2, u_2, 0} D (ComplexShape.up ℤ)) (A := ℤ) 1)
      (Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) K)))
  ((CategoryTheory.Functor.map (G.mapHomologicalComplex (ComplexShape.up ℤ)) (triangle φ).mor₃ ≫
        NatTrans.app (Iso.hom (Functor.commShiftIso (G.mapHomologicalComplex (ComplexShape.up ℤ)) 1)) K) ≫
      CategoryTheory.Functor.map
        (CategoryTheory.shiftFunctor (HomologicalComplex.{v_2, u_2, 0} D (ComplexShape.up ℤ)) (A := ℤ) 1)
        (𝟙 (Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) K)) :
    Quiver.Hom (V := HomologicalComplex.{v_2, u_2, 0} D (ComplexShape.up ℤ))
      (Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) (mappingCone φ))
      (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v_2, u_2, 0} D (ComplexShape.up ℤ)) (A := ℤ) 1)
        (Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) K)))
  (CategoryStruct.comp (Y :=
      mappingCone (F := Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) K)
        (CategoryTheory.Functor.map (G.mapHomologicalComplex (ComplexShape.up ℤ)) φ))
      (Iso.hom (mapHomologicalComplexIso φ G) :
        Quiver.Hom (V := HomologicalComplex.{v_2, u_2, 0} D (ComplexShape.up ℤ))
          (Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) (mappingCone φ))
          (mappingCone (F := Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) K)
            (CategoryTheory.Functor.map (G.mapHomologicalComplex (ComplexShape.up ℤ)) φ)))
      (Triangle.mor₃
          (triangle (K := Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) K)
            (CategoryTheory.Functor.map (G.mapHomologicalComplex (ComplexShape.up ℤ)) φ)) :
        Quiver.Hom (V := HomologicalComplex.{v_2, u_2, 0} D (ComplexShape.up ℤ))
          (Triangle.obj₃
            (triangle (K := Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) K)
              (CategoryTheory.Functor.map (G.mapHomologicalComplex (ComplexShape.up ℤ)) φ)))
          (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v_2, u_2, 0} D (ComplexShape.up ℤ)) (A := ℤ) 1)
            (Triangle.obj₁
              (triangle (K := Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) K)
                (CategoryTheory.Functor.map (G.mapHomologicalComplex (ComplexShape.up ℤ)) φ))))) :
    Quiver.Hom (V := HomologicalComplex.{v_2, u_2, 0} D (ComplexShape.up ℤ))
      (Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) (mappingCone φ))
      (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v_2, u_2, 0} D (ComplexShape.up ℤ)) (A := ℤ) 1)
        (Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) K)))
Tactic: dsimp