-- Module: Mathlib.CategoryTheory.Galois.Prorepresentability
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
inst_1 : GaloisCategory.{u₁, u₂} C
F : C ⥤ FintypeCat
x x_1 : (PointedGaloisObject F)ᵒᵖ
A : C
a : (F.obj A).obj
isGalois : IsGalois.{u₁, u₂} A
B : C
b : (F.obj B).obj
isGalois_1 : IsGalois.{u₁, u₂} B
x_2 :
  Opposite.op { obj := A, pt := a, isGalois := isGalois : PointedGaloisObject F } ⟶
    Opposite.op { obj := B, pt := b, isGalois := isGalois_1 }
f :
  (Opposite.unop (Opposite.op { obj := B, pt := b, isGalois := isGalois_1 : PointedGaloisObject F })).obj ⟶
    (Opposite.unop (Opposite.op { obj := A, pt := a, isGalois := isGalois : PointedGaloisObject F })).obj
hf :
  (@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))
          _ _ _ (F.map f) :
        (a_1 :
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
              (F.obj (Opposite.unop (Opposite.op { obj := B, pt := b, isGalois := isGalois_1 })).obj)) →
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
            (F.obj (Opposite.unop (Opposite.op { obj := A, pt := a, isGalois := isGalois })).obj))
      b =
    a
w :
  ∀ (Y : C) (g : A ⟶ Y),
    Eq (α :=
      (Functor.obj (Functor.obj (const (PointedGaloisObject F)ᵒᵖ) (F ⋙ FintypeCat.incl))
            (Opposite.op { obj := B, pt := b, isGalois := isGalois_1 : PointedGaloisObject F })).obj
        Y)
      (TypeCat.Fun.toFun
        (ConcreteCategory.hom (FC := TypeCat.Fun)
          (NatTrans.app
            (map (Functor.op (incl F) ⋙ coyoneda)
                (Opposite.op
                  { val := f, comp := hf :
                    (Opposite.unop
                          (Opposite.op { obj := B, pt := b, isGalois := isGalois_1 : PointedGaloisObject F })).Hom
                      (Opposite.unop
                        (Opposite.op { obj := A, pt := a, isGalois := isGalois : PointedGaloisObject F })) }) ≫
              match (motive :=
                (x : (PointedGaloisObject F)ᵒᵖ) →
                  Functor.obj (Functor.op (incl F) ⋙ coyoneda) x ⟶
                    Functor.obj (Functor.obj (const (PointedGaloisObject F)ᵒᵖ) (F ⋙ FintypeCat.incl)) x)
                Opposite.op { obj := B, pt := b, isGalois := isGalois_1 : PointedGaloisObject F } with
              | Opposite.op { obj := A, pt := a, isGalois := isGalois } =>
                {
                  app := fun (X : C) ↦
                    ↾fun (f : A ⟶ X) ↦
                      (@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))
                            _ _ _ (F.map f) :
                          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
                            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
                        a,
                  naturality := ⋯ :
                  NatTrans
                    (Functor.obj (Functor.op (incl F) ⋙ coyoneda)
                      (Opposite.op { obj := A, pt := a, isGalois := isGalois : PointedGaloisObject F }))
                    (Functor.obj (Functor.obj (const (PointedGaloisObject F)ᵒᵖ) (F ⋙ FintypeCat.incl))
                      (Opposite.op { obj := A, pt := a, isGalois := isGalois : PointedGaloisObject F })) })
            Y))
        g)
      (TypeCat.Fun.toFun
        (ConcreteCategory.hom (FC := TypeCat.Fun)
          (NatTrans.app
            (CategoryStruct.comp (Y :=
              Functor.obj (Functor.obj (const (PointedGaloisObject F)ᵒᵖ) (F ⋙ FintypeCat.incl))
                (Opposite.op { obj := A, pt := a, isGalois := isGalois : PointedGaloisObject F }))
              (match (motive :=
                (x : (PointedGaloisObject F)ᵒᵖ) →
                  Functor.obj (Functor.op (incl F) ⋙ coyoneda) x ⟶
                    Functor.obj (Functor.obj (const (PointedGaloisObject F)ᵒᵖ) (F ⋙ FintypeCat.incl)) x)
                Opposite.op { obj := A, pt := a, isGalois := isGalois : PointedGaloisObject F } with
              | Opposite.op { obj := A, pt := a, isGalois := isGalois } =>
                {
                  app := fun (X : C) ↦
                    ↾fun (f : A ⟶ X) ↦
                      (@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))
                            _ _ _ (F.map f) :
                          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
                            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
                        a,
                  naturality := ⋯ :
                  NatTrans
                    (Functor.obj (Functor.op (incl F) ⋙ coyoneda)
                      (Opposite.op { obj := A, pt := a, isGalois := isGalois : PointedGaloisObject F }))
                    (Functor.obj (Functor.obj (const (PointedGaloisObject F)ᵒᵖ) (F ⋙ FintypeCat.incl))
                      (Opposite.op { obj := A, pt := a, isGalois := isGalois : PointedGaloisObject F })) })
              (map (Functor.obj (const (PointedGaloisObject F)ᵒᵖ) (F ⋙ FintypeCat.incl))
                (Opposite.op
                  { val := f, comp := hf :
                    (Opposite.unop
                          (Opposite.op { obj := B, pt := b, isGalois := isGalois_1 : PointedGaloisObject F })).Hom
                      (Opposite.unop
                        (Opposite.op { obj := A, pt := a, isGalois := isGalois : PointedGaloisObject F })) })))
            Y))
        g)
⊢ Eq (α :=
    Functor.obj (Functor.op (incl F) ⋙ coyoneda)
        (Opposite.op { obj := A, pt := a, isGalois := isGalois : PointedGaloisObject F }) ⟶
      Functor.obj (Functor.obj (const (PointedGaloisObject F)ᵒᵖ) (F ⋙ FintypeCat.incl))
        (Opposite.op { obj := B, pt := b, isGalois := isGalois_1 : PointedGaloisObject F }))
    (map (Functor.op (incl F) ⋙ coyoneda)
        (Opposite.op
          { val := f, comp := hf :
            (Opposite.unop (Opposite.op { obj := B, pt := b, isGalois := isGalois_1 : PointedGaloisObject F })).Hom
              (Opposite.unop (Opposite.op { obj := A, pt := a, isGalois := isGalois : PointedGaloisObject F })) }) ≫
      match (motive :=
        (x : (PointedGaloisObject F)ᵒᵖ) →
          Functor.obj (Functor.op (incl F) ⋙ coyoneda) x ⟶
            Functor.obj (Functor.obj (const (PointedGaloisObject F)ᵒᵖ) (F ⋙ FintypeCat.incl)) x)
        Opposite.op { obj := B, pt := b, isGalois := isGalois_1 : PointedGaloisObject F } with
      | Opposite.op { obj := A, pt := a, isGalois := isGalois } =>
        {
          app := fun (X : C) ↦
            ↾fun (f : A ⟶ X) ↦
              (@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))
                    _ _ _ (F.map f) :
                  (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
                    (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
                a,
          naturality := ⋯ })
    (CategoryStruct.comp (Y :=
      Functor.obj (Functor.obj (const (PointedGaloisObject F)ᵒᵖ) (F ⋙ FintypeCat.incl))
        (Opposite.op { obj := A, pt := a, isGalois := isGalois : PointedGaloisObject F }))
      (match (motive :=
        (x : (PointedGaloisObject F)ᵒᵖ) →
          Functor.obj (Functor.op (incl F) ⋙ coyoneda) x ⟶
            Functor.obj (Functor.obj (const (PointedGaloisObject F)ᵒᵖ) (F ⋙ FintypeCat.incl)) x)
        Opposite.op { obj := A, pt := a, isGalois := isGalois : PointedGaloisObject F } with
      | Opposite.op { obj := A, pt := a, isGalois := isGalois } =>
        {
          app := fun (X : C) ↦
            ↾fun (f : A ⟶ X) ↦
              (@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))
                    _ _ _ (F.map f) :
                  (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
                    (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
                a,
          naturality := ⋯ })
      (map (Functor.obj (const (PointedGaloisObject F)ᵒᵖ) (F ⋙ FintypeCat.incl))
        (Opposite.op { val := f, comp := hf })))

Tactic:
ext Y (g : (A : C) ⟶ Y)