-- 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
E : Type u_2
inst_4 : Category.{v₁, u_2} E
inst_5 : Preadditive.{v₁, u_2} E
F : Functor.{v₁, v₁, u₁, u_1} C D
inst_6 : F.Additive
G : Functor.{v₁, v₁, u_1, u_2} D E
inst_7 : G.Additive
M N : Mat_ C
f : Quiver.Hom.{v₁, max 1 u₁} M N
i : (obj.{v₁, v₁, max 1 u₁, max 1 u_2} (F ⋙ G).mapMat_ M).ι
j : (obj.{v₁, v₁, max 1 u₁, max 1 u_2} (comp.{v₁, v₁, v₁, max 1 u₁, max 1 u_1, max 1 u_2} F.mapMat_ G.mapMat_) N).ι
⊢ Eq (α :=
    (fun (i : (obj.{v₁, v₁, max 1 u₁, max 1 u_2} (F ⋙ G).mapMat_ M).ι)
          (j :
            (obj.{v₁, v₁, max 1 u₁, max 1 u_2} (comp.{v₁, v₁, v₁, max 1 u₁, max 1 u_1, max 1 u_2} F.mapMat_ G.mapMat_)
                N).ι) ↦
        Quiver.Hom.{v₁, u_2} ((obj.{v₁, v₁, max 1 u₁, max 1 u_2} (F ⋙ G).mapMat_ M).X i)
          ((obj.{v₁, v₁, max 1 u₁, max 1 u_2} (comp.{v₁, v₁, v₁, max 1 u₁, max 1 u_1, max 1 u_2} F.mapMat_ G.mapMat_)
                N).X
            j))
      i j)
    (CategoryStruct.comp.{v₁, max 1 u_2} (map.{v₁, v₁, max 1 u₁, max 1 u_2} (F ⋙ G).mapMat_ f)
        (Iso.hom.{v₁, max 1 u_2}
          ((fun (M : Mat_ C) ↦
              (eqToIso.{v₁, max 1 u_2} ⋯ :
                Iso.{v₁, max 1 u_2} (obj.{v₁, v₁, max 1 u₁, max 1 u_2} (F ⋙ G).mapMat_ M)
                  (obj.{v₁, v₁, max 1 u₁, max 1 u_2}
                    (comp.{v₁, v₁, v₁, max 1 u₁, max 1 u_1, max 1 u_2} F.mapMat_ G.mapMat_) M)))
            N))
        i j :
      (fun (i : (obj.{v₁, v₁, max 1 u₁, max 1 u_2} (F ⋙ G).mapMat_ M).ι)
            (j :
              (obj.{v₁, v₁, max 1 u₁, max 1 u_2} (comp.{v₁, v₁, v₁, max 1 u₁, max 1 u_1, max 1 u_2} F.mapMat_ G.mapMat_)
                  N).ι) ↦
          Quiver.Hom.{v₁, u_2} ((obj.{v₁, v₁, max 1 u₁, max 1 u_2} (F ⋙ G).mapMat_ M).X i)
            ((obj.{v₁, v₁, max 1 u₁, max 1 u_2} (comp.{v₁, v₁, v₁, max 1 u₁, max 1 u_1, max 1 u_2} F.mapMat_ G.mapMat_)
                  N).X
              j))
        i j)
    (CategoryStruct.comp.{v₁, max 1 u_2}
        (Iso.hom.{v₁, max 1 u_2}
          ((fun (M : Mat_ C) ↦
              (eqToIso.{v₁, max 1 u_2} ⋯ :
                Iso.{v₁, max 1 u_2} (obj.{v₁, v₁, max 1 u₁, max 1 u_2} (F ⋙ G).mapMat_ M)
                  (obj.{v₁, v₁, max 1 u₁, max 1 u_2}
                    (comp.{v₁, v₁, v₁, max 1 u₁, max 1 u_1, max 1 u_2} F.mapMat_ G.mapMat_) M)))
            M))
        (map.{v₁, v₁, max 1 u₁, max 1 u_2} (comp.{v₁, v₁, v₁, max 1 u₁, max 1 u_1, max 1 u_2} F.mapMat_ G.mapMat_) f) i
        j :
      (fun (i : (obj.{v₁, v₁, max 1 u₁, max 1 u_2} (F ⋙ G).mapMat_ M).ι)
            (j :
              (obj.{v₁, v₁, max 1 u₁, max 1 u_2} (comp.{v₁, v₁, v₁, max 1 u₁, max 1 u_1, max 1 u_2} F.mapMat_ G.mapMat_)
                  N).ι) ↦
          Quiver.Hom.{v₁, u_2} ((obj.{v₁, v₁, max 1 u₁, max 1 u_2} (F ⋙ G).mapMat_ M).X i)
            ((obj.{v₁, v₁, max 1 u₁, max 1 u_2} (comp.{v₁, v₁, v₁, max 1 u₁, max 1 u_1, max 1 u_2} F.mapMat_ G.mapMat_)
                  N).X
              j))
        i j)
have H : ∀ (ι : Type) [fintype : Fintype ι] (X : ι → C)
  (f : Quiver.Hom.{v₁, max 1 u₁} { ι := ι, fintype := fintype, X := X : Mat_ C } N)
  (i : (obj.{v₁, v₁, max 1 u₁, max 1 u_2} (F ⋙ G).mapMat_ { ι := ι, fintype := fintype, X := X : Mat_ C }).ι),
  Eq (α :=
    (fun (i : (obj.{v₁, v₁, max 1 u₁, max 1 u_2} (F ⋙ G).mapMat_ { ι := ι, fintype := fintype, X := X : Mat_ C }).ι)
          (j :
            (obj.{v₁, v₁, max 1 u₁, max 1 u_2} (comp.{v₁, v₁, v₁, max 1 u₁, max 1 u_1, max 1 u_2} F.mapMat_ G.mapMat_)
                N).ι) ↦
        Quiver.Hom.{v₁, u_2}
          ((obj.{v₁, v₁, max 1 u₁, max 1 u_2} (F ⋙ G).mapMat_ { ι := ι, fintype := fintype, X := X : Mat_ C }).X i)
          ((obj.{v₁, v₁, max 1 u₁, max 1 u_2} (comp.{v₁, v₁, v₁, max 1 u₁, max 1 u_1, max 1 u_2} F.mapMat_ G.mapMat_)
                N).X
            j))
      i j)
    (CategoryStruct.comp.{v₁, max 1 u_2} (map.{v₁, v₁, max 1 u₁, max 1 u_2} (F ⋙ G).mapMat_ f)
        (Iso.hom.{v₁, max 1 u_2}
          ((fun (M : Mat_ C) ↦
              (eqToIso.{v₁, max 1 u_2} sorry :
                Iso.{v₁, max 1 u_2} (obj.{v₁, v₁, max 1 u₁, max 1 u_2} (F ⋙ G).mapMat_ M)
                  (obj.{v₁, v₁, max 1 u₁, max 1 u_2}
                    (comp.{v₁, v₁, v₁, max 1 u₁, max 1 u_1, max 1 u_2} F.mapMat_ G.mapMat_) M)))
            N))
        i j :
      (fun (i : (obj.{v₁, v₁, max 1 u₁, max 1 u_2} (F ⋙ G).mapMat_ { ι := ι, fintype := fintype, X := X : Mat_ C }).ι)
            (j :
              (obj.{v₁, v₁, max 1 u₁, max 1 u_2} (comp.{v₁, v₁, v₁, max 1 u₁, max 1 u_1, max 1 u_2} F.mapMat_ G.mapMat_)
                  N).ι) ↦
          Quiver.Hom.{v₁, u_2}
            ((obj.{v₁, v₁, max 1 u₁, max 1 u_2} (F ⋙ G).mapMat_ { ι := ι, fintype := fintype, X := X : Mat_ C }).X i)
            ((obj.{v₁, v₁, max 1 u₁, max 1 u_2} (comp.{v₁, v₁, v₁, max 1 u₁, max 1 u_1, max 1 u_2} F.mapMat_ G.mapMat_)
                  N).X
              j))
        i j)
    (CategoryStruct.comp.{v₁, max 1 u_2}
        (Iso.hom.{v₁, max 1 u_2}
          ((fun (M : Mat_ C) ↦
              (eqToIso.{v₁, max 1 u_2} sorry :
                Iso.{v₁, max 1 u_2} (obj.{v₁, v₁, max 1 u₁, max 1 u_2} (F ⋙ G).mapMat_ M)
                  (obj.{v₁, v₁, max 1 u₁, max 1 u_2}
                    (comp.{v₁, v₁, v₁, max 1 u₁, max 1 u_1, max 1 u_2} F.mapMat_ G.mapMat_) M)))
            { ι := ι, fintype := fintype, X := X }))
        (map.{v₁, v₁, max 1 u₁, max 1 u_2} (comp.{v₁, v₁, v₁, max 1 u₁, max 1 u_1, max 1 u_2} F.mapMat_ G.mapMat_) f) i
        j :
      (fun (i : (obj.{v₁, v₁, max 1 u₁, max 1 u_2} (F ⋙ G).mapMat_ { ι := ι, fintype := fintype, X := X : Mat_ C }).ι)
            (j :
              (obj.{v₁, v₁, max 1 u₁, max 1 u_2} (comp.{v₁, v₁, v₁, max 1 u₁, max 1 u_1, max 1 u_2} F.mapMat_ G.mapMat_)
                  N).ι) ↦
          Quiver.Hom.{v₁, u_2}
            ((obj.{v₁, v₁, max 1 u₁, max 1 u_2} (F ⋙ G).mapMat_ { ι := ι, fintype := fintype, X := X : Mat_ C }).X i)
            ((obj.{v₁, v₁, max 1 u₁, max 1 u_2} (comp.{v₁, v₁, v₁, max 1 u₁, max 1 u_1, max 1 u_2} F.mapMat_ G.mapMat_)
                  N).X
              j))
        i j)
Tactic: cases M
Cleared hypotheses: M, f, i