-- Lean 4 proof step from Mathlib.CategoryTheory.Preadditive.Mat (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : Preadditive.{v₁, u₁} C
D : Type u₁
inst_2 : Category.{v₁, u₁} D
inst_3 : Preadditive.{v₁, u₁} D
inst_4 : HasFiniteBiproducts.{v₁, u₁} D
F : Functor.{v₁, v₁, max 1 u₁, u₁} (Mat_ C) D
inst_5 : F.Additive
M N : Mat_ C
f : Quiver.Hom.{v₁, max 1 u₁} M N
⊢ Eq (α :=
    F.obj M ⟶ biproduct.{0, v₁, u₁} fun (i : N.ι) ↦ F.obj (Functor.obj.{v₁, v₁, u₁, max 1 u₁} (embedding C) (N.X i)))
    (F.map f ≫ Iso.hom (additiveObjIsoBiproduct F N) :
      F.obj M ⟶ biproduct.{0, v₁, u₁} fun (i : N.ι) ↦ F.obj (Functor.obj.{v₁, v₁, u₁, max 1 u₁} (embedding C) (N.X i)))
    (CategoryStruct.comp (Y :=
        biproduct.{0, v₁, u₁} fun (i : M.ι) ↦ F.obj (Functor.obj.{v₁, v₁, u₁, max 1 u₁} (embedding C) (M.X i)))
        (Iso.hom (additiveObjIsoBiproduct F M) :
          F.obj M ⟶
            biproduct.{0, v₁, u₁} fun (i : M.ι) ↦ F.obj (Functor.obj.{v₁, v₁, u₁, max 1 u₁} (embedding C) (M.X i)))
        (biproduct.matrix fun (i : M.ι) (j : N.ι) ↦ F.map (Functor.map (embedding C) (f i j)) :
          Quiver.Hom.{v₁, u₁} (V := D)
            (biproduct.{0, v₁, u₁} fun (i : M.ι) ↦ F.obj (Functor.obj.{v₁, v₁, u₁, max 1 u₁} (embedding C) (M.X i)))
            (biproduct.{0, v₁, u₁} fun (j : N.ι) ↦ F.obj (Functor.obj.{v₁, v₁, u₁, max 1 u₁} (embedding C) (N.X j)))) :
      F.obj M ⟶ biproduct.{0, v₁, u₁} fun (j : N.ι) ↦ F.obj (Functor.obj.{v₁, v₁, u₁, max 1 u₁} (embedding C) (N.X j)))
have w : ∀ (i : N.ι),
  CategoryStruct.comp (Y :=
      biproduct.{0, v₁, u₁} fun (i : N.ι) ↦ F.obj (Functor.obj.{v₁, v₁, u₁, max 1 u₁} (embedding C) (N.X i)))
      (F.map f ≫ Iso.hom (additiveObjIsoBiproduct F N) :
        F.obj M ⟶
          biproduct.{0, v₁, u₁} fun (i : N.ι) ↦ F.obj (Functor.obj.{v₁, v₁, u₁, max 1 u₁} (embedding C) (N.X i)))
      (biproduct.π (fun (i : N.ι) ↦ F.obj (Functor.obj.{v₁, v₁, u₁, max 1 u₁} (embedding C) (N.X i))) i :
        (biproduct.{0, v₁, u₁} fun (i : N.ι) ↦ F.obj (Functor.obj.{v₁, v₁, u₁, max 1 u₁} (embedding C) (N.X i))) ⟶
          F.obj (Functor.obj.{v₁, v₁, u₁, max 1 u₁} (embedding C) (N.X i))) =
    CategoryStruct.comp (Y :=
      biproduct.{0, v₁, u₁} fun (i : N.ι) ↦ F.obj (Functor.obj.{v₁, v₁, u₁, max 1 u₁} (embedding C) (N.X i)))
      (CategoryStruct.comp (Y :=
          biproduct.{0, v₁, u₁} fun (i : M.ι) ↦ F.obj (Functor.obj.{v₁, v₁, u₁, max 1 u₁} (embedding C) (M.X i)))
          (Iso.hom (additiveObjIsoBiproduct F M) :
            F.obj M ⟶
              biproduct.{0, v₁, u₁} fun (i : M.ι) ↦ F.obj (Functor.obj.{v₁, v₁, u₁, max 1 u₁} (embedding C) (M.X i)))
          (biproduct.matrix fun (i : M.ι) (j : N.ι) ↦ F.map (Functor.map (embedding C) (f i j)) :
            Quiver.Hom.{v₁, u₁} (V := D)
              (biproduct.{0, v₁, u₁} fun (i : M.ι) ↦ F.obj (Functor.obj.{v₁, v₁, u₁, max 1 u₁} (embedding C) (M.X i)))
              (biproduct.{0, v₁, u₁} fun (j : N.ι) ↦
                F.obj (Functor.obj.{v₁, v₁, u₁, max 1 u₁} (embedding C) (N.X j)))) :
        F.obj M ⟶
          biproduct.{0, v₁, u₁} fun (j : N.ι) ↦ F.obj (Functor.obj.{v₁, v₁, u₁, max 1 u₁} (embedding C) (N.X j)))
      (biproduct.π (fun (i : N.ι) ↦ F.obj (Functor.obj.{v₁, v₁, u₁, max 1 u₁} (embedding C) (N.X i))) i :
        (biproduct.{0, v₁, u₁} fun (i : N.ι) ↦ F.obj (Functor.obj.{v₁, v₁, u₁, max 1 u₁} (embedding C) (N.X i))) ⟶
          F.obj (Functor.obj.{v₁, v₁, u₁, max 1 u₁} (embedding C) (N.X i)))
Tactic: classical
  ext i : 1
  simp only [Category.assoc, additiveObjIsoBiproduct_hom_π, isoBiproductEmbedding_hom,
    embedding_obj_ι, embedding_obj_X, biproduct.lift_π, biproduct.matrix_π,
    ← cancel_epi (additiveObjIsoBiproduct F M).inv, Iso.inv_hom_id_assoc]
  ext j : 1
  simp only [ι_additiveObjIsoBiproduct_inv_assoc, isoBiproductEmbedding_inv,
    biproduct.ι_desc, ← F.map_comp]
  congr 1
  funext ⟨⟩ ⟨⟩
  simp [comp_apply, dite_comp, comp_dite]