-- Module: Mathlib.CategoryTheory.Galois.Prorepresentability
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
inst_1 : GaloisCategory.{u₁, u₂} C
F : C ⥤ FintypeCat
inst_2 : FiberFunctor F
X 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
u : A ⟶ X
v : B ⟶ X
h :
  Eq (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj 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 u) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
        a :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj 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 v) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj B)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
        b :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
Z : C
z : (F.obj Z).obj
isGalois_2 : IsGalois.{u₁, u₂} Z
f :
  (Opposite.unop (Opposite.op { obj := Z, pt := z, isGalois := isGalois_2 : PointedGaloisObject F })).obj ⟶
    (Opposite.unop (Opposite.op { obj := A, pt := a, 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 := Z, pt := z, isGalois := isGalois_2 })).obj)) →
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
            (F.obj (Opposite.unop (Opposite.op { obj := A, pt := a, isGalois := ⋯ })).obj))
      (Opposite.unop (Opposite.op { obj := Z, pt := z, isGalois := isGalois_2 : PointedGaloisObject F })).pt =
    (Opposite.unop (Opposite.op { obj := A, pt := a, isGalois := ⋯ : PointedGaloisObject F })).pt
h_1 : True
g :
  (Opposite.unop (Opposite.op { obj := Z, pt := z, isGalois := isGalois_2 : PointedGaloisObject F })).obj ⟶
    (Opposite.unop (Opposite.op { obj := B, pt := b, isGalois := ⋯ : PointedGaloisObject F })).obj
hg :
  (@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 (Opposite.unop (Opposite.op { obj := Z, pt := z, isGalois := isGalois_2 })).obj)) →
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
            (F.obj (Opposite.unop (Opposite.op { obj := B, pt := b, isGalois := ⋯ })).obj))
      (Opposite.unop (Opposite.op { obj := Z, pt := z, isGalois := isGalois_2 : PointedGaloisObject F })).pt =
    (Opposite.unop (Opposite.op { obj := B, pt := b, isGalois := ⋯ : PointedGaloisObject F })).pt
refine_2 :
  Eq.{u₂ + 1} (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
    ((fun (f : Z ⟶ 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 Z)) →
                (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
            z :
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (map ((Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X)
              (Opposite.op
                { val := f, comp := hf :
                  (Opposite.unop (Opposite.op { obj := Z, pt := z, isGalois := ⋯ : PointedGaloisObject F })).Hom
                    (Opposite.unop
                      (Opposite.op { obj := A, pt := a, isGalois := isGalois : PointedGaloisObject F })) })) :
          (a :
              (fun (X : Type u₂) ↦ X)
                ((((incl F).op ⋙ coyoneda) ⋙ (evaluation C (Type u₂)).obj X).obj
                  (Opposite.op { obj := A, pt := a, isGalois := isGalois }))) →
            (fun (X : Type u₂) ↦ X)
              ((((incl F).op ⋙ coyoneda) ⋙ (evaluation C (Type u₂)).obj X).obj
                (Opposite.op { obj := Z, pt := z, isGalois := ⋯ })))
        u))
    ((fun (f : Z ⟶ 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 Z)) →
                (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
            z :
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (map ((Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X)
              (Opposite.op
                { val := g, comp := hg :
                  (Opposite.unop (Opposite.op { obj := Z, pt := z, isGalois := ⋯ : PointedGaloisObject F })).Hom
                    (Opposite.unop
                      (Opposite.op { obj := B, pt := b, isGalois := isGalois_1 : PointedGaloisObject F })) })) :
          (a :
              (fun (X : Type u₂) ↦ X)
                ((((incl F).op ⋙ coyoneda) ⋙ (evaluation C (Type u₂)).obj X).obj
                  (Opposite.op { obj := B, pt := b, isGalois := isGalois_1 }))) →
            (fun (X : Type u₂) ↦ X)
              ((((incl F).op ⋙ coyoneda) ⋙ (evaluation C (Type u₂)).obj X).obj
                (Opposite.op { obj := Z, pt := z, isGalois := ⋯ })))
        v))
⊢ Eq (α :=
    (fun (X : Type u₂) ↦ X)
      (Functor.obj ((Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X)
        (Opposite.op { obj := Z, pt := z, isGalois := ⋯ : PointedGaloisObject F })))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (map ((Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X)
              (Opposite.op
                { val := f, comp := hf :
                  (Opposite.unop (Opposite.op { obj := Z, pt := z, isGalois := ⋯ : PointedGaloisObject F })).Hom
                    (Opposite.unop
                      (Opposite.op { obj := A, pt := a, isGalois := isGalois : PointedGaloisObject F })) })) :
          (a :
              (fun (X : Type u₂) ↦ X)
                ((((incl F).op ⋙ coyoneda) ⋙ (evaluation C (Type u₂)).obj X).obj
                  (Opposite.op { obj := A, pt := a, isGalois := isGalois }))) →
            (fun (X : Type u₂) ↦ X)
              ((((incl F).op ⋙ coyoneda) ⋙ (evaluation C (Type u₂)).obj X).obj
                (Opposite.op { obj := Z, pt := z, isGalois := ⋯ })))
        u :
      (fun (X : Type u₂) ↦ X)
        (Functor.obj ((Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X)
          (Opposite.op { obj := Z, pt := z, isGalois := ⋯ : PointedGaloisObject F })))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (map ((Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X)
              (Opposite.op
                { val := g, comp := hg :
                  (Opposite.unop (Opposite.op { obj := Z, pt := z, isGalois := ⋯ : PointedGaloisObject F })).Hom
                    (Opposite.unop
                      (Opposite.op { obj := B, pt := b, isGalois := isGalois_1 : PointedGaloisObject F })) })) :
          (a :
              (fun (X : Type u₂) ↦ X)
                ((((incl F).op ⋙ coyoneda) ⋙ (evaluation C (Type u₂)).obj X).obj
                  (Opposite.op { obj := B, pt := b, isGalois := isGalois_1 }))) →
            (fun (X : Type u₂) ↦ X)
              ((((incl F).op ⋙ coyoneda) ⋙ (evaluation C (Type u₂)).obj X).obj
                (Opposite.op { obj := Z, pt := z, isGalois := ⋯ })))
        v :
      (fun (X : Type u₂) ↦ X)
        (Functor.obj ((Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X)
          (Opposite.op { obj := Z, pt := z, isGalois := ⋯ : PointedGaloisObject F })))

Tactic:
apply evaluation_injective_of_isConnected F Z X z

Premises used: CategoryTheory.PreGaloisCategory.evaluation_injective_of_isConnected