-- 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))))
    (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.inv (F.mapId { as := op (X (α j)) : LocallyDiscrete Cᵒᵖ })))
            (D.obj (α j)) ≫
          D.hom (𝟙 (X (α j)) ≫ f (α j)) (𝟙 (X (α j))) (q' (α j) ≫ p' (β (α j))) ⋯ ⋯ ≫
            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 : 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))) sorry sorry) ≫
      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))))))
        (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))))) ≫
          pullFunctorObjHom w D (p' j ≫ q' (α j) ≫ f' (β (α j))) (p' j ≫ q' (α j)) (𝟙 (X' j)) sorry sorry ≫
            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))))
Tactic: simp only [Category.id_comp, Functor.map_comp, Category.assoc]