-- Module: Mathlib.CategoryTheory.Galois.Decomposition
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
inst_1 : GaloisCategory.{u₁, u₂} C
F : Functor.{u₂, w, u₁, w + 1} C FintypeCat
inst_2 : FiberFunctor F
X A : C
u : A ⟶ selfProd F X
a : (F.obj A).obj
h :
  Eq (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F 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 (selfProd F X)))
        a :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
    (mkSelfProdFib F X : (F.obj (selfProd F X)).obj)
inst_3 : IsConnected.{u₁, u₂} A
b : (F.obj A).obj
t : (F.obj X).obj
h₁ :
  Eq (α := (F.obj X).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 (selfProdProj u t)) :
          (a : (F.obj A).obj) → (F.obj X).obj)
        b :
      (F.obj X).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 (Pi.π (fun (x : (F.obj X).obj) ↦ X) t) :
              F.obj (piObj.{w, u₂, u₁} fun (x : (F.obj X).obj) ↦ X) ⟶ F.obj X) :
          (a : (F.obj (∏ᶜ fun (x : (F.obj X).obj) ↦ X)).obj) → (F.obj X).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
                    (Pi.map' (⇑(fiberPerm h b) : (a : (F.obj X).obj) → (F.obj X).obj) fun (j : (F.obj X).obj) ↦
                      Iso.hom (Iso.refl X)) :
                  F.obj (selfProd F X) ⟶ F.obj (selfProd F X)) :
              (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X))) →
                (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X)))
            (mkSelfProdFib F X) :
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj (selfProd F X))) :
      (F.obj X).obj)
⊢ Eq (α := (F.obj X).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 (selfProdProj u t)) :
          (a : (F.obj A).obj) → (F.obj X).obj)
        b :
      (F.obj X).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 (Pi.π (fun (x : (F.obj X).obj) ↦ X) t) :
              F.obj (piObj.{w, u₂, u₁} fun (x : (F.obj X).obj) ↦ X) ⟶ F.obj X) :
          (a : (F.obj (∏ᶜ fun (x : (F.obj X).obj) ↦ X)).obj) → (F.obj X).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
                    (u ≫
                      Pi.map' (⇑(fiberPerm h b) : (a : (F.obj X).obj) → (F.obj X).obj) fun (j : (F.obj X).obj) ↦
                        Iso.hom (Iso.refl X)) :
                  F.obj A ⟶ F.obj (selfProd F X)) :
              (a : (F.obj A).obj) → (F.obj (selfProd F X)).obj)
            a :
          (F.obj (selfProd F X)).obj) :
      (F.obj X).obj)

Tactic:
rw [map_comp, FintypeCat.comp_apply, h]

Premises used: FintypeCat.comp_apply, CategoryTheory.Functor.map_comp