-- 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
Y : C
g : A ⟶ Y
w :
  Eq (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj Y))
    ((@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 g) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj Y))
        ((@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 :
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
            (F.obj
              (Opposite.unop (Opposite.op { obj := A, pt := a, isGalois := isGalois : PointedGaloisObject F })).obj)) :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj Y))
    ((@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 g) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj Y))
        a :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj 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)

Tactic:
suffices h : F.map g (F.map f b) = F.map g a by simpa