-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.Basic (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
K : GrothendieckTopology.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
ℱ : Functor.{v_2, v, u_2, v + 1} Dᵒᵖ (Type v)
ℱ' : Sheaf K (Type v)
α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.obj (P := Presheaf.IsSheaf K)
inst_2 : G.IsLocallyFull K
X : D
x : ℱ.obj (op X)
Y : C
f : G.obj Y ⟶ X
l : CoverByImageStructure G f
⊢ Eq (α := ℱ'.obj (P := Presheaf.IsSheaf K).obj (op (G.obj Y)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (ℱ'.obj (P := Presheaf.IsSheaf K).map l.lift.op) :
          (a : ℱ'.obj.obj (op (G.obj l.1))) → ℱ'.obj.obj (op (G.obj Y)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (α.app (op l.1)) :
              (a : (G.op ⋙ ℱ).obj (op l.1)) → (G.op ⋙ ℱ'.obj).obj (op l.1))
            ((ConcreteCategory.hom (FC := TypeCat.Fun) (ℱ.map l.map.op) : (a : ℱ.obj (op X)) → ℱ.obj (op (G.obj l.1)))
                x :
              ℱ.obj (op (G.obj l.1))) :
          (G.op ⋙ ℱ'.obj (P := Presheaf.IsSheaf K)).obj (op l.1)) :
      ℱ'.obj (P := Presheaf.IsSheaf K).obj (op (G.obj Y)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (α.app (op Y)) :
          (a : (G.op ⋙ ℱ).obj (op Y)) → (G.op ⋙ ℱ'.obj).obj (op Y))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (ℱ.map f.op) : (a : ℱ.obj (op X)) → ℱ.obj (op (G.obj Y))) x :
          ℱ.obj (op (G.obj Y))) :
      (G.op ⋙ ℱ'.obj (P := Presheaf.IsSheaf K)).obj (op Y))
have obtain : ∀ (W : C) (iYW : G.obj Y ⟶ G.obj W) (iWX : G.obj W ⟶ X),
  Eq (α := ℱ'.obj (P := Presheaf.IsSheaf K).obj (op (G.obj Y)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (ℱ'.obj (P := Presheaf.IsSheaf K).map
              { obj := W, lift := iYW, map := iWX, fac := sorry : CoverByImageStructure G (iYW ≫ iWX) }.lift.op) :
          (a : ℱ'.obj.obj (op (G.obj { obj := W, lift := iYW, map := iWX, fac := sorry }.1))) → ℱ'.obj.obj (op (G.obj Y)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (α.app (op { obj := W, lift := iYW, map := iWX, fac := sorry : CoverByImageStructure G (iYW ≫ iWX) }.1)) :
              (a : (G.op ⋙ ℱ).obj (op { obj := W, lift := iYW, map := iWX, fac := sorry }.1)) →
                (G.op ⋙ ℱ'.obj).obj (op { obj := W, lift := iYW, map := iWX, fac := sorry }.1))
            ((ConcreteCategory.hom (FC := TypeCat.Fun)
                    (ℱ.map
                      { obj := W, lift := iYW, map := iWX, fac := sorry : CoverByImageStructure G (iYW ≫ iWX) }.map.op) :
                  (a : ℱ.obj (op X)) → ℱ.obj (op (G.obj { obj := W, lift := iYW, map := iWX, fac := sorry }.1)))
                x :
              ℱ.obj (op (G.obj { obj := W, lift := iYW, map := iWX, fac := sorry }.1))) :
          (G.op ⋙ ℱ'.obj (P := Presheaf.IsSheaf K)).obj (op { obj := W, lift := iYW, map := iWX, fac := sorry }.1)) :
      ℱ'.obj (P := Presheaf.IsSheaf K).obj (op (G.obj Y)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (α.app (op Y)) :
          (a : (G.op ⋙ ℱ).obj (op Y)) → (G.op ⋙ ℱ'.obj).obj (op Y))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (ℱ.map (iYW ≫ iWX).op) : (a : ℱ.obj (op X)) → ℱ.obj (op (G.obj Y)))
            x :
          ℱ.obj (op (G.obj Y))) :
      (G.op ⋙ ℱ'.obj (P := Presheaf.IsSheaf K)).obj (op Y))
Tactic: obtain ⟨W, iYW, iWX, rfl⟩ := l
Cleared hypotheses: f, l