-- 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 u_1
inst_2 : Category.{v₁, u_1} D
inst_3 : Preadditive.{v₁, u_1} D
M N : Mat_ C
f : Quiver.Hom.{v₁, max 1 u₁} M N
⊢ Eq (α :=
    Quiver.Hom.{v₁, max 1 u₁} (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (mapMat_ (𝟭 C)) M)
      (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (Functor.id.{v₁, max 1 u₁} (Mat_ C)) N))
    (CategoryStruct.comp.{v₁, max 1 u₁} (map.{v₁, v₁, max 1 u₁, max 1 u₁} (mapMat_ (𝟭 C)) f)
      (Iso.hom.{v₁, max 1 u₁}
        ((fun (M : Mat_ C) ↦
            (eqToIso.{v₁, max 1 u₁} ⋯ :
              Iso.{v₁, max 1 u₁} (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (mapMat_ (𝟭 C)) M)
                (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (Functor.id.{v₁, max 1 u₁} (Mat_ C)) M)))
          N)))
    (CategoryStruct.comp.{v₁, max 1 u₁}
      (Iso.hom.{v₁, max 1 u₁}
        ((fun (M : Mat_ C) ↦
            (eqToIso.{v₁, max 1 u₁} ⋯ :
              Iso.{v₁, max 1 u₁} (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (mapMat_ (𝟭 C)) M)
                (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (Functor.id.{v₁, max 1 u₁} (Mat_ C)) M)))
          M))
      (map.{v₁, v₁, max 1 u₁, max 1 u₁} (Functor.id.{v₁, max 1 u₁} (Mat_ C)) f))
have H : ∀ (i : (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (mapMat_ (𝟭 C)) M).ι)
  (j : (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (Functor.id.{v₁, max 1 u₁} (Mat_ C)) N).ι),
  Eq (α :=
    (fun (i : (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (mapMat_ (𝟭 C)) M).ι)
          (j : (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (Functor.id.{v₁, max 1 u₁} (Mat_ C)) N).ι) ↦
        Quiver.Hom.{v₁, u₁} ((obj.{v₁, v₁, max 1 u₁, max 1 u₁} (mapMat_ (𝟭 C)) M).X i)
          ((obj.{v₁, v₁, max 1 u₁, max 1 u₁} (Functor.id.{v₁, max 1 u₁} (Mat_ C)) N).X j))
      i j)
    (CategoryStruct.comp.{v₁, max 1 u₁} (map.{v₁, v₁, max 1 u₁, max 1 u₁} (mapMat_ (𝟭 C)) f)
        (Iso.hom.{v₁, max 1 u₁}
          ((fun (M : Mat_ C) ↦
              (eqToIso.{v₁, max 1 u₁} sorry :
                Iso.{v₁, max 1 u₁} (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (mapMat_ (𝟭 C)) M)
                  (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (Functor.id.{v₁, max 1 u₁} (Mat_ C)) M)))
            N))
        i j :
      (fun (i : (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (mapMat_ (𝟭 C)) M).ι)
            (j : (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (Functor.id.{v₁, max 1 u₁} (Mat_ C)) N).ι) ↦
          Quiver.Hom.{v₁, u₁} ((obj.{v₁, v₁, max 1 u₁, max 1 u₁} (mapMat_ (𝟭 C)) M).X i)
            ((obj.{v₁, v₁, max 1 u₁, max 1 u₁} (Functor.id.{v₁, max 1 u₁} (Mat_ C)) N).X j))
        i j)
    (CategoryStruct.comp.{v₁, max 1 u₁}
        (Iso.hom.{v₁, max 1 u₁}
          ((fun (M : Mat_ C) ↦
              (eqToIso.{v₁, max 1 u₁} sorry :
                Iso.{v₁, max 1 u₁} (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (mapMat_ (𝟭 C)) M)
                  (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (Functor.id.{v₁, max 1 u₁} (Mat_ C)) M)))
            M))
        (map.{v₁, v₁, max 1 u₁, max 1 u₁} (Functor.id.{v₁, max 1 u₁} (Mat_ C)) f) i j :
      (fun (i : (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (mapMat_ (𝟭 C)) M).ι)
            (j : (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (Functor.id.{v₁, max 1 u₁} (Mat_ C)) N).ι) ↦
          Quiver.Hom.{v₁, u₁} ((obj.{v₁, v₁, max 1 u₁, max 1 u₁} (mapMat_ (𝟭 C)) M).X i)
            ((obj.{v₁, v₁, max 1 u₁, max 1 u₁} (Functor.id.{v₁, max 1 u₁} (Mat_ C)) N).X j))
        i j)
Tactic: classical
    ext
    cases M; cases N
    simp [comp_dite, dite_comp]