-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Descent.DescentData (goal)
Goal:
C : Type u
inst : Category.{v, u} C
F : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
S' : C
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S'
e : S' ≅ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j ≫ e.hom
β : ι → ι'
q' : (i : ι) → X i ⟶ X' (β i)
w' : ∀ (i : ι), q' i ≫ f' (β i) = f i ≫ e.inv
D : F.DescentData f
j : ι'
⊢ Eq (α :=
    Quiver.Hom (V := (↑(F.obj { as := op (X' j) : LocallyDiscrete Cᵒᵖ }) : Type u'))
      (Functor.obj (C := (↑(F.obj { as := op (X (α j)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
        (Cat.Hom.toFunctor (F.map (p' j).op.toLoc)) (D.obj (α j)))
      (Functor.obj (C := (↑(F.obj { as := op (X (α j)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
        (Cat.Hom.toFunctor (F.map (p' j).op.toLoc)) (D.obj (α j))))
    (Functor.map (Cat.Hom.toFunctor (F.map (p' j).op.toLoc))
        (NatTrans.app (Cat.Hom₂.toNatTrans (Iso.inv (F.mapId { as := op (X (α j)) : LocallyDiscrete Cᵒᵖ })))
          (D.obj (α j))) ≫
      Functor.map (Cat.Hom.toFunctor (F.map (p' j).op.toLoc))
          (D.hom (f (α j)) (𝟙 (X (α j))) (q' (α j) ≫ p' (β (α j))) ⋯ ⋯) ≫
        CategoryStruct.comp (Y :=
          Functor.obj (C := (↑(F.obj { as := op (X (α j)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
            (Cat.Hom.toFunctor (F.map (p' j).op.toLoc))
            (Functor.obj (C := (↑(F.obj { as := op (X' (β (α j))) : LocallyDiscrete Cᵒᵖ }) : Type u'))
              (Cat.Hom.toFunctor (F.map (Quiver.Hom.toLoc (Quiver.Hom.op (q' (α j))))))
              (Functor.obj (C := (↑(F.obj { as := op (X (α (β (α j)))) : LocallyDiscrete Cᵒᵖ }) : Type u'))
                (Cat.Hom.toFunctor (F.map (Quiver.Hom.toLoc (Quiver.Hom.op (p' (β (α j))))))) (D.obj (α (β (α j)))))))
          (Functor.map (Cat.Hom.toFunctor (F.map (p' j).op.toLoc))
            (NatTrans.app
              (Cat.Hom₂.toNatTrans
                (Iso.hom
                  (F.mapComp' (Quiver.Hom.toLoc (Quiver.Hom.op (p' (β (α j)))))
                    (Quiver.Hom.toLoc (Quiver.Hom.op (q' (α j))))
                    (CategoryStruct.comp (Y := { as := op (X' (β (α j))) })
                      (Quiver.Hom.toLoc (Quiver.Hom.op (p' (β (α j))))) (Quiver.Hom.toLoc (Quiver.Hom.op (q' (α j)))))
                    ⋯)))
              (D.obj (α (β (α j))))))
          (NatTrans.app
              (Cat.Hom₂.toNatTrans
                (Iso.inv
                  (F.mapComp' (Quiver.Hom.toLoc (Quiver.Hom.op (q' (α j)))) (p' j).op.toLoc
                    (CategoryStruct.comp (Y := { as := op (X (α j)) }) (Quiver.Hom.toLoc (Quiver.Hom.op (q' (α j))))
                      (p' j).op.toLoc)
                    ⋯)))
              (Functor.obj (C := (↑(F.obj { as := op (X (α (β (α j)))) : LocallyDiscrete Cᵒᵖ }) : Type u'))
                (Cat.Hom.toFunctor (F.map (Quiver.Hom.toLoc (Quiver.Hom.op (p' (β (α j))))))) (D.obj (α (β (α j))))) ≫
            pullFunctorObjHom w D (p' j ≫ q' (α j) ≫ f' (β (α j))) (p' j ≫ q' (α j)) (𝟙 (X' j)) ⋯ ⋯ ≫
              NatTrans.app (Cat.Hom₂.toNatTrans (Iso.hom (F.mapId { as := op (X' j) : LocallyDiscrete Cᵒᵖ })))
                (Functor.obj (C := (↑(F.obj { as := op (X (α j)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
                  (Cat.Hom.toFunctor (F.map (p' j).op.toLoc)) (D.obj (α j)))))
    (𝟙
      (Functor.obj (C := (↑(F.obj { as := op (X (α j)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
        (Cat.Hom.toFunctor (F.map (p' j).op.toLoc)) (D.obj (α j))))
have h : Functor.map (Cat.Hom.toFunctor (F.map (p' j).op.toLoc))
      (NatTrans.app (Cat.Hom₂.toNatTrans (Iso.inv (F.mapId { as := op (X (α j)) : LocallyDiscrete Cᵒᵖ })))
        (D.obj (α j))) ≫
    (F.mapComp' (Quiver.Hom.toLoc (Quiver.Hom.op (𝟙 (X (α j))))) (p' j).op.toLoc (p' j).op.toLoc sorry).inv.toNatTrans.app
        (D.obj (α j)) ≫
      D.hom (p' j ≫ f (α j)) (p' j) (p' j ≫ q' (α j) ≫ p' (β (α j))) sorry sorry ≫
        (F.mapComp' (Quiver.Hom.toLoc (Quiver.Hom.op (q' (α j) ≫ p' (β (α j))))) (p' j).op.toLoc
                    (p' j ≫ q' (α j) ≫ p' (β (α j))).op.toLoc sorry).hom.toNatTrans.app
            (D.obj (α (β (α j)))) ≫
          CategoryStruct.comp (Y :=
            Functor.obj (C := (↑(F.obj { as := op (X (α j)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
              (Cat.Hom.toFunctor (F.map (p' j).op.toLoc))
              (Functor.obj (C := (↑(F.obj { as := op (X' (β (α j))) : LocallyDiscrete Cᵒᵖ }) : Type u'))
                (Cat.Hom.toFunctor (F.map (Quiver.Hom.toLoc (Quiver.Hom.op (q' (α j))))))
                (Functor.obj (C := (↑(F.obj { as := op (X (α (β (α j)))) : LocallyDiscrete Cᵒᵖ }) : Type u'))
                  (Cat.Hom.toFunctor (F.map (Quiver.Hom.toLoc (Quiver.Hom.op (p' (β (α j))))))) (D.obj (α (β (α j)))))))
            (Functor.map (Cat.Hom.toFunctor (F.map (p' j).op.toLoc))
              (NatTrans.app
                (Cat.Hom₂.toNatTrans
                  (Iso.hom
                    (F.mapComp' (Quiver.Hom.toLoc (Quiver.Hom.op (p' (β (α j)))))
                      (Quiver.Hom.toLoc (Quiver.Hom.op (q' (α j))))
                      (CategoryStruct.comp (Y := { as := op (X' (β (α j))) })
                        (Quiver.Hom.toLoc (Quiver.Hom.op (p' (β (α j))))) (Quiver.Hom.toLoc (Quiver.Hom.op (q' (α j)))))
                      sorry)))
                (D.obj (α (β (α j))))))
            (CategoryStruct.comp (Y :=
              Functor.obj (C := (↑(F.obj { as := op (X' (β (α j))) : LocallyDiscrete Cᵒᵖ }) : Type u'))
                (Cat.Hom.toFunctor
                  (F.map
                    (CategoryStruct.comp (Y := { as := op (X (α j)) }) (Quiver.Hom.toLoc (Quiver.Hom.op (q' (α j))))
                      (p' j).op.toLoc)))
                (Functor.obj (C := (↑(F.obj { as := op (X (α (β (α j)))) : LocallyDiscrete Cᵒᵖ }) : Type u'))
                  (Cat.Hom.toFunctor (F.map (Quiver.Hom.toLoc (Quiver.Hom.op (p' (β (α j))))))) (D.obj (α (β (α j))))))
              (NatTrans.app
                  (Cat.Hom₂.toNatTrans
                    (Iso.inv
                      (F.mapComp' (Quiver.Hom.toLoc (Quiver.Hom.op (q' (α j)))) (p' j).op.toLoc
                        (CategoryStruct.comp (Y := { as := op (X (α j)) }) (Quiver.Hom.toLoc (Quiver.Hom.op (q' (α j))))
                          (p' j).op.toLoc)
                        sorry)))
                  (Functor.obj (C := (↑(F.obj { as := op (X (α (β (α j)))) : LocallyDiscrete Cᵒᵖ }) : Type u'))
                    (Cat.Hom.toFunctor (F.map (Quiver.Hom.toLoc (Quiver.Hom.op (p' (β (α j)))))))
                    (D.obj (α (β (α j))))) :
                Quiver.Hom (V := (↑(F.obj { as := op (X' j) : LocallyDiscrete Cᵒᵖ }) : Type u'))
                  (Functor.obj (C := (↑(F.obj { as := op (X' (β (α j))) : LocallyDiscrete Cᵒᵖ }) : Type u'))
                    (Functor.comp (D := (↑(F.obj { as := op (X (α j)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
                      (Cat.Hom.toFunctor (F.map (Quiver.Hom.toLoc (Quiver.Hom.op (q' (α j))))))
                      (Cat.Hom.toFunctor (F.map (p' j).op.toLoc)))
                    (Functor.obj (C := (↑(F.obj { as := op (X (α (β (α j)))) : LocallyDiscrete Cᵒᵖ }) : Type u'))
                      (Cat.Hom.toFunctor (F.map (Quiver.Hom.toLoc (Quiver.Hom.op (p' (β (α j)))))))
                      (D.obj (α (β (α j))))))
                  (Functor.obj (C := (↑(F.obj { as := op (X' (β (α j))) : LocallyDiscrete Cᵒᵖ }) : Type u'))
                    (Cat.Hom.toFunctor
                      (F.map
                        (CategoryStruct.comp (Y := { as := op (X (α j)) }) (Quiver.Hom.toLoc (Quiver.Hom.op (q' (α j))))
                          (p' j).op.toLoc)))
                    (Functor.obj (C := (↑(F.obj { as := op (X (α (β (α j)))) : LocallyDiscrete Cᵒᵖ }) : Type u'))
                      (Cat.Hom.toFunctor (F.map (Quiver.Hom.toLoc (Quiver.Hom.op (p' (β (α j)))))))
                      (D.obj (α (β (α j)))))))
              ((F.mapComp' (Quiver.Hom.toLoc (Quiver.Hom.op (p' (β (α j))))) (p' j ≫ q' (α j)).op.toLoc
                            (p' j ≫ q' (α j) ≫ p' (β (α j))).op.toLoc sorry).inv.toNatTrans.app
                    (D.obj (α (β (α j)))) ≫
                  D.hom (p' j ≫ f (α j)) (p' j ≫ q' (α j) ≫ p' (β (α j))) (p' j) sorry sorry ≫
                    (F.mapComp' (p' j).op.toLoc (Quiver.Hom.toLoc (Quiver.Hom.op (𝟙 (X' j)))) (p' j).op.toLoc
                                sorry).hom.toNatTrans.app
                        (D.obj (α j)) ≫
                      NatTrans.app (Cat.Hom₂.toNatTrans (Iso.hom (F.mapId { as := op (X' j) })))
                        (Functor.obj (C := (↑(F.obj { as := op (X (α j)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
                          (Cat.Hom.toFunctor (F.map (p' j).op.toLoc)) (D.obj (α j))) :
                Quiver.Hom (V := (↑(F.obj { as := op (X' j) : LocallyDiscrete Cᵒᵖ }) : Type u'))
                  (Functor.obj (C := (↑(F.obj { as := op (X' (β (α j))) : LocallyDiscrete Cᵒᵖ }) : Type u'))
                    (Cat.Hom.toFunctor (F.map (p' j ≫ q' (α j)).op.toLoc))
                    (Functor.obj (C := (↑(F.obj { as := op (X (α (β (α j)))) : LocallyDiscrete Cᵒᵖ }) : Type u'))
                      (Cat.Hom.toFunctor (F.map (Quiver.Hom.toLoc (Quiver.Hom.op (p' (β (α j)))))))
                      (D.obj (α (β (α j))))))
                  (Functor.obj (C := (↑(F.obj { as := op (X (α j)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
                    (Cat.Hom.toFunctor (F.map (p' j).op.toLoc)) (D.obj (α j))))) =
  𝟙
    (Functor.obj (C := (↑(F.obj { as := op (X (α j)) : LocallyDiscrete Cᵒᵖ }) : Type u'))
      (Cat.Hom.toFunctor (F.map (p' j).op.toLoc)) (D.obj (α j)))
Tactic: rw [pullFunctorObjHom_eq_assoc _ _ _ _ _ (p' _ ≫ f _) (p' _ ≫ q' _ ≫ p' _) (p' _) (by simp)
        (by simp [w', reassoc_of% w]),
      map_eq_pullHom_assoc _ (p' j) (p' j) (p' _ ≫ q' _ ≫ p' _) (by simp) (by simp),
      D.pullHom_hom _ _ (p' j ≫ f _) (by simp) _ _ (by simp)
        (by simp [w, reassoc_of% w']) _ _ (by simp) rfl]