-- Lean 4 proof step from Mathlib.CategoryTheory.Action.Concrete (goal)
Goal:
G : Type u_1
inst : Group G
H N : Subgroup G
inst_1 : Fintype (G ⧸ N)
inst_2 : Fintype (G ⧸ H)
h : N ≤ H
g : G
⊢ Eq.{u_1 + 1} (α := V (ofMulAction G (FintypeCat.of (G ⧸ N))) ⟶ V (ofMulAction G (FintypeCat.of (G ⧸ H))))
    (CategoryStruct.comp (Y := V (ofMulAction G (FintypeCat.of (G ⧸ N))))
      ((ρ (ofMulAction G (FintypeCat.of (G ⧸ N))) : (a : G) → End (ofMulAction G (FintypeCat.of (G ⧸ N))).V) g)
      (FintypeCat.homMk (Quotient.lift (fun (x : G) ↦ ⟦x⟧) ⋯)))
    (CategoryStruct.comp (Y := V (ofMulAction G (FintypeCat.of (G ⧸ H))))
      (FintypeCat.homMk (Quotient.lift (fun (x : G) ↦ ⟦x⟧) ⋯))
      ((ρ (ofMulAction G (FintypeCat.of (G ⧸ H))) : (a : G) → End (ofMulAction G (FintypeCat.of (G ⧸ H))).V) g))
have h₁ : ∀ (x : G ⧸ N),
  Eq (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (V (ofMulAction G (FintypeCat.of (G ⧸ H)))))
    ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
              (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
            _ _ _
            (CategoryStruct.comp (Y := V (ofMulAction G (FintypeCat.of (G ⧸ N))))
                ((ρ (ofMulAction G (FintypeCat.of (G ⧸ N))) : (a : G) → End (ofMulAction G (FintypeCat.of (G ⧸ N))).V)
                  g)
                (FintypeCat.homMk (Quotient.lift (fun (x : G) ↦ ⟦x⟧) sorry)) :
              V (ofMulAction G (FintypeCat.of (G ⧸ N))) ⟶ V (ofMulAction G (FintypeCat.of (G ⧸ H)))) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (ofMulAction G (FintypeCat.of (G ⧸ N))).V) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (ofMulAction G (FintypeCat.of (G ⧸ H))).V)
        x :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (V (ofMulAction G (FintypeCat.of (G ⧸ H)))))
    ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
              (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
            _ _ _
            (CategoryStruct.comp (Y := V (ofMulAction G (FintypeCat.of (G ⧸ H))))
                (FintypeCat.homMk (Quotient.lift (fun (x : G) ↦ ⟦x⟧) sorry))
                ((ρ (ofMulAction G (FintypeCat.of (G ⧸ H))) : (a : G) → End (ofMulAction G (FintypeCat.of (G ⧸ H))).V)
                  g) :
              V (ofMulAction G (FintypeCat.of (G ⧸ N))) ⟶ V (ofMulAction G (FintypeCat.of (G ⧸ H)))) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (ofMulAction G (FintypeCat.of (G ⧸ N))).V) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (ofMulAction G (FintypeCat.of (G ⧸ H))).V)
        x :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (V (ofMulAction G (FintypeCat.of (G ⧸ H)))))
Tactic: ext (x : G ⧸ N)