-- Module: Mathlib.CategoryTheory.Galois.Basic
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
inst_1 : PreGaloisCategory.{u₁, u₂} C
inst_2 : FiberFunctor F
X Y : C
f g : X ⟶ Y
x : (F.obj X).obj
h :
  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 f) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj Y))
        x :
      (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 X)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj Y))
        x :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj Y))
simp :
  (@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 (equalizer.ι f g) : F.obj (equalizer f g) ⟶ F.obj X) :
        (a : (F.obj (equalizer f g)).obj) → (F.obj X).obj)
      ((Equiv.symm
              (Equiv.trans (Iso.toEquiv (PreservesEqualizer.iso (F ⋙ FintypeCat.incl) f g))
                (Iso.toEquiv
                  (Types.equalizerIso ((F.map f).hom (F := ObjectProperty.FullSubcategory.obj))
                    ((F.map g).hom (F := ObjectProperty.FullSubcategory.obj))))) :
            (a :
                { x : (F.obj X).obj //
                  (ConcreteCategory.hom (F.map f) : (a : (F.obj X).obj) → (F.obj Y).obj) x =
                    (ConcreteCategory.hom (F.map g) : (a : (F.obj X).obj) → (F.obj Y).obj) x }) →
              (F.obj (equalizer f g)).obj)
          (Subtype.mk (p := fun (x : (F.obj X).obj) ↦
            Eq (α := (F.obj Y).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.map f) :
                    (a : (F.obj X).obj) → (F.obj Y).obj)
                  x :
                (F.obj Y).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.map g) :
                    (a : (F.obj X).obj) → (F.obj Y).obj)
                  x :
                (F.obj Y).obj))
            x h) :
        (F.obj (equalizer f g)).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 (equalizer.ι f g) : F.obj (equalizer f g) ⟶ F.obj X) :
        (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (equalizer f g))) →
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
      (((Equiv.symm (fiberEqualizerEquiv F f g) :
              { x : (F.obj X).obj //
                  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 f) :
                          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X)) →
                            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj Y))
                        x :
                      (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 X)) →
                            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj Y))
                        x :
                      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj Y)) } ≃
                (F.obj (equalizer f g)).obj) :
            (a :
                { x : (F.obj X).obj //
                  (ConcreteCategory.hom (F.map f) :
                        (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X)) →
                          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj Y))
                      x =
                    (ConcreteCategory.hom (F.map g) :
                        (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X)) →
                          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj Y))
                      x }) →
              (F.obj (equalizer f g)).obj)
          (Subtype.mk (p := fun (x : (F.obj X).obj) ↦
            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 f) :
                    (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X)) →
                      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj Y))
                  x :
                (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 X)) →
                      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj Y))
                  x :
                (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj Y)))
            x h) :
        (F.obj (equalizer f g)).obj) =
    x

Tactic:
simp only [fiberEqualizerEquiv, comp_obj, FintypeCat.incl_obj, Functor.comp_map, Iso.toEquiv_comp]

Premises used: CategoryTheory.Functor.comp_obj, CategoryTheory.PreGaloisCategory.fiberEqualizerEquiv, CategoryTheory.Iso.toEquiv_comp, FintypeCat.incl_obj, CategoryTheory.Functor.comp_map