-- Module: Mathlib.CategoryTheory.Preadditive.Mat
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
ι : Type
fintype : Fintype ι
X : ι → C
i : (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (mapMat_ (𝟭 C)) { ι := ι, fintype := fintype, X := X : Mat_ C }).ι
ι_1 : Type
fintype_1 : Fintype ι_1
X_1 : ι_1 → C
j :
  (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (Functor.id.{v₁, max 1 u₁} (Mat_ C))
      { ι := ι_1, fintype := fintype_1, X := X_1 : Mat_ C }).ι
f :
  Quiver.Hom.{v₁, max 1 u₁} { ι := ι, fintype := fintype, X := X : Mat_ C } { ι := ι_1, fintype := fintype_1, X := X_1 }
⊢ Eq (α :=
    (fun (i : (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (mapMat_ (𝟭 C)) { ι := ι, fintype := fintype, X := X : Mat_ C }).ι)
          (j :
            (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (Functor.id.{v₁, max 1 u₁} (Mat_ C))
                { ι := ι_1, fintype := fintype_1, X := X_1 : Mat_ C }).ι) ↦
        Quiver.Hom.{v₁, u₁}
          ((obj.{v₁, v₁, max 1 u₁, max 1 u₁} (mapMat_ (𝟭 C)) { ι := ι, fintype := fintype, X := X : Mat_ C }).X i)
          ((obj.{v₁, v₁, max 1 u₁, max 1 u₁} (Functor.id.{v₁, max 1 u₁} (Mat_ C))
                { ι := ι_1, fintype := fintype_1, X := X_1 : Mat_ C }).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} ⋯ :
                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)))
            { ι := ι_1, fintype := fintype_1, X := X_1 }))
        i j :
      (fun (i : (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (mapMat_ (𝟭 C)) { ι := ι, fintype := fintype, X := X : Mat_ C }).ι)
            (j :
              (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (Functor.id.{v₁, max 1 u₁} (Mat_ C))
                  { ι := ι_1, fintype := fintype_1, X := X_1 : Mat_ C }).ι) ↦
          Quiver.Hom.{v₁, u₁}
            ((obj.{v₁, v₁, max 1 u₁, max 1 u₁} (mapMat_ (𝟭 C)) { ι := ι, fintype := fintype, X := X : Mat_ C }).X i)
            ((obj.{v₁, v₁, max 1 u₁, max 1 u₁} (Functor.id.{v₁, max 1 u₁} (Mat_ C))
                  { ι := ι_1, fintype := fintype_1, X := X_1 : Mat_ C }).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} ⋯ :
                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)))
            { ι := ι, fintype := fintype, X := X }))
        (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)) { ι := ι, fintype := fintype, X := X : Mat_ C }).ι)
            (j :
              (obj.{v₁, v₁, max 1 u₁, max 1 u₁} (Functor.id.{v₁, max 1 u₁} (Mat_ C))
                  { ι := ι_1, fintype := fintype_1, X := X_1 : Mat_ C }).ι) ↦
          Quiver.Hom.{v₁, u₁}
            ((obj.{v₁, v₁, max 1 u₁, max 1 u₁} (mapMat_ (𝟭 C)) { ι := ι, fintype := fintype, X := X : Mat_ C }).X i)
            ((obj.{v₁, v₁, max 1 u₁, max 1 u₁} (Functor.id.{v₁, max 1 u₁} (Mat_ C))
                  { ι := ι_1, fintype := fintype_1, X := X_1 : Mat_ C }).X
              j))
        i j)

Tactic:
simp [comp_dite, dite_comp]

Premises used: CategoryTheory.dite_comp, CategoryTheory.comp_dite