-- 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.obj₂ (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.mor₂ (Functor.obj (Functor.mapTriangle (G.mapHomologicalComplex (ComplexShape.up ℤ))) (triangle φ)) ≫
        Iso.hom (mapHomologicalComplexIso φ G) :
      Quiver.Hom (V := HomologicalComplex.{v_2, u_2, 0} D (ComplexShape.up ℤ))
        (Triangle.obj₂ (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))))
    (Iso.hom
          (Iso.refl
            (Triangle.obj₂
              (Functor.obj (Functor.mapTriangle (G.mapHomologicalComplex (ComplexShape.up ℤ))) (triangle φ)))) ≫
        Triangle.mor₂
          (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.obj₂ (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))))
have refine_2 : Eq (α :=
  Quiver.Hom (V := HomologicalComplex.{v_2, u_2, 0} D (ComplexShape.up ℤ))
    (Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) L)
    (mappingCone (F := Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) K)
      (CategoryTheory.Functor.map (G.mapHomologicalComplex (ComplexShape.up ℤ)) φ)))
  (CategoryStruct.comp (Y := Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) (mappingCone φ))
      (CategoryTheory.Functor.map (G.mapHomologicalComplex (ComplexShape.up ℤ)) (inr 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)) L)
      (mappingCone (F := Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) K)
        (CategoryTheory.Functor.map (G.mapHomologicalComplex (ComplexShape.up ℤ)) φ)))
  (𝟙 (Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) L) ≫
      inr (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)) L)
      (mappingCone (F := Functor.obj (G.mapHomologicalComplex (ComplexShape.up ℤ)) K)
        (CategoryTheory.Functor.map (G.mapHomologicalComplex (ComplexShape.up ℤ)) φ)))
Tactic: dsimp