-- 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
f : End F
A : PointedGaloisObject F
change :
  Eq (α :=
    (fun (X : Type u₂) ↦ X)
      ((Functor.obj (Functor.obj (whiskeringRight C FintypeCat (Type u₂)) FintypeCat.incl) F).obj A.obj))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (NatTrans.app
              ((FullyFaithful.homEquiv ((FullyFaithful.ofFullyFaithful FintypeCat.incl).whiskeringRight C) :
                  (a : F ⟶ F) →
                    ((whiskeringRight C FintypeCat (Type u₂)).obj FintypeCat.incl).obj F ⟶
                      ((whiskeringRight C FintypeCat (Type u₂)).obj FintypeCat.incl).obj F)
                f)
              A.obj) :
          (a :
              (fun (X : Type u₂) ↦ X)
                ((((whiskeringRight C FintypeCat (Type u₂)).obj FintypeCat.incl).obj F).obj A.obj)) →
            (fun (X : Type u₂) ↦ X) ((((whiskeringRight C FintypeCat (Type u₂)).obj FintypeCat.incl).obj F).obj A.obj))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (NatTrans.app
                  (CategoryStruct.comp (Y := colimit (J := (PointedGaloisObject F)ᵒᵖ) (Functor.op (incl F) ⋙ coyoneda))
                    (colimit.ι (Functor.op (incl F) ⋙ coyoneda) (Opposite.op A))
                    (Iso.hom
                      (colimit.isoColimitCocone
                        { cocone := cocone F, isColimit := isColimit F :
                          ColimitCocone (J := (PointedGaloisObject F)ᵒᵖ) (Functor.op (incl F) ⋙ coyoneda) })))
                  A.obj) :
              (a : (fun (X : Type u₂) ↦ X) ((((incl F).op ⋙ coyoneda).obj (Opposite.op A)).obj A.obj)) →
                (fun (X : Type u₂) ↦ X) ({ cocone := cocone F, isColimit := isColimit F }.cocone.pt.obj A.obj))
            (𝟙 (Functor.obj (incl F) A)) :
          (fun (X : Type u₂) ↦ X)
            ((Cocone.pt (J := (PointedGaloisObject F)ᵒᵖ)
                  (ColimitCocone.cocone
                    { cocone := cocone F, isColimit := isColimit F :
                      ColimitCocone (J := (PointedGaloisObject F)ᵒᵖ) (Functor.op (incl F) ⋙ coyoneda) })).obj
              A.obj)) :
      (fun (X : Type u₂) ↦ X)
        ((Functor.obj (Functor.obj (whiskeringRight C FintypeCat (Type u₂)) FintypeCat.incl) F).obj A.obj))
    ((@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.app A.obj : F.obj A.obj ⟶ F.obj A.obj) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A.obj)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A.obj))
        A.pt :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A.obj))
⊢ Eq (α := (F.obj A.obj).obj)
    (ULift.down.{u₁, u₂}
        {
          down :=
            (ConcreteCategory.hom (FC := TypeCat.Fun)
                  (NatTrans.app
                    (CategoryStruct.comp (Y := Cocone.pt (J := (PointedGaloisObject F)ᵒᵖ) (cocone F))
                      (Iso.hom
                          (colimit.isoColimitCocone
                            { cocone := cocone F, isColimit := isColimit F :
                              ColimitCocone (J := (PointedGaloisObject F)ᵒᵖ) (Functor.op (incl F) ⋙ coyoneda) }) :
                        colimit (J := (PointedGaloisObject F)ᵒᵖ) (Functor.op (incl F) ⋙ coyoneda) ⟶
                          Cocone.pt (J := (PointedGaloisObject F)ᵒᵖ) (cocone F))
                      ((FullyFaithful.homEquiv ((FullyFaithful.ofFullyFaithful FintypeCat.incl).whiskeringRight C) :
                          (a : End F) → End (F ⋙ FintypeCat.incl))
                        f))
                    (Functor.obj (incl F) A)) :
                (a : (colimit ((incl F).op ⋙ coyoneda)).obj ((incl F).obj A)) →
                  (F ⋙ FintypeCat.incl).obj ((incl F).obj A))
              ((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
                      (NatTrans.app (colimit.ι (Functor.op (incl F) ⋙ coyoneda) (Opposite.op A))
                        (Functor.obj (incl F) A)) :
                    (a : (incl F).obj A ⟶ (incl F).obj A) → (colimit ((incl F).op ⋙ coyoneda)).obj ((incl F).obj A))
                  (𝟙 (Functor.obj (incl F) A)) :
                (colimit (J := (PointedGaloisObject F)ᵒᵖ) (Functor.op (incl F) ⋙ coyoneda)).obj
                  (Functor.obj (incl F) A)) } :
      (F.obj A.obj).obj)
    ((@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.app A.obj : F.obj A.obj ⟶ F.obj A.obj) :
          (a : (F.obj A.obj).obj) → (F.obj A.obj).obj)
        A.pt :
      (F.obj A.obj).obj)

Tactic:
change (((FullyFaithful.whiskeringRight (FullyFaithful.ofFullyFaithful
      FintypeCat.incl) C).homEquiv) f).app A
    (((colimit.ι _ _) ≫ (colimit.isoColimitCocone ⟨cocone F, isColimit F⟩).hom).app
      A _) = f.app A A.pt

Premises used: CategoryTheory.PreGaloisCategory.PointedGaloisObject.pt, CategoryTheory.Limits.colimit.isoColimitCocone, CategoryTheory.ConcreteCategory.hom, CategoryTheory.Functor.FullyFaithful.homEquiv, CategoryTheory.PreGaloisCategory.PointedGaloisObject.obj, CategoryTheory.PreGaloisCategory.PointedGaloisObject.isColimit, CategoryTheory.Limits.ColimitCocone.mk, FintypeCat.incl, CategoryTheory.CategoryStruct.comp, CategoryTheory.PreGaloisCategory.PointedGaloisObject.cocone, CategoryTheory.Iso.hom, DFunLike.coe, CategoryTheory.Functor.FullyFaithful.ofFullyFaithful, CategoryTheory.Functor.FullyFaithful.whiskeringRight, CategoryTheory.NatTrans.app, CategoryTheory.Limits.colimit.ι