-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Sites.Representability (goal)
Goal:
F : Sheaf zariskiTopology (Type u)
ι : Type u
X : ι → Scheme
f : (i : ι) → Functor.obj yoneda (X i) ⟶ F.obj (P := Presheaf.IsSheaf zariskiTopology)
hf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)
i_1 j_1 k : ι
i j : (glueData hf).J
a :
  Eq (α := Functor.obj yoneda (⋯.pullback (f j)) ⟶ F.obj (P := Presheaf.IsSheaf zariskiTopology))
    ((Equiv.symm yonedaEquiv : (a : F.obj.obj (op (⋯.pullback (f j)))) → yoneda.obj (⋯.pullback (f j)) ⟶ F.obj)
        ((yonedaEquiv : (a : yoneda.obj (⋯.pullback (f j)) ⟶ F.obj) → F.obj.obj (op (⋯.pullback (f j))))
            (CategoryTheory.Functor.map yoneda (⋯.fst' (f j)) ≫ f i) :
          F.obj (P := Presheaf.IsSheaf zariskiTopology).obj (op (⋯.pullback (f j)))) :
      Functor.obj yoneda (⋯.pullback (f j)) ⟶ F.obj (P := Presheaf.IsSheaf zariskiTopology))
    ((Equiv.symm yonedaEquiv : (a : F.obj.obj (op (⋯.pullback (f j)))) → yoneda.obj (⋯.pullback (f j)) ⟶ F.obj)
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (F.obj (P := Presheaf.IsSheaf zariskiTopology).map ((⋯.fst' (f i)).op ≫ (⋯.symmetry ⋯).op)) :
              (a : F.obj.obj (op (X j))) → F.obj.obj (op (⋯.pullback (f j))))
            ((yonedaEquiv : (a : yoneda.obj (X j) ⟶ F.obj) → F.obj.obj (op (X j))) (f j) :
              F.obj (P := Presheaf.IsSheaf zariskiTopology).obj (op (X j))) :
          F.obj (P := Presheaf.IsSheaf zariskiTopology).obj (op (⋯.pullback (f j)))) :
      Functor.obj yoneda (⋯.pullback (f j)) ⟶ F.obj (P := Presheaf.IsSheaf zariskiTopology))
a₁ :
  CategoryTheory.Functor.map yoneda (⋯.fst' (f j)) ≫ f i =
    (Equiv.symm yonedaEquiv : (a : F.obj.obj (op (⋯.pullback (f j)))) → yoneda.obj (⋯.pullback (f j)) ⟶ F.obj)
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
              (F.obj (P := Presheaf.IsSheaf zariskiTopology).map ((⋯.fst' (f i)).op ≫ (⋯.symmetry ⋯).op)) :
            (a : F.obj.obj (op (X j))) → F.obj.obj (op (⋯.pullback (f j))))
          ((yonedaEquiv : (a : yoneda.obj (X j) ⟶ F.obj) → F.obj.obj (op (X j))) (f j) :
            F.obj (P := Presheaf.IsSheaf zariskiTopology).obj (op (X j))) :
        F.obj (P := Presheaf.IsSheaf zariskiTopology).obj (op (⋯.pullback (f j))))
⊢ Eq (α := Functor.obj yoneda (⋯.pullback (f j)) ⟶ F.obj (P := Presheaf.IsSheaf zariskiTopology))
    ((Equiv.symm yonedaEquiv : (a : F.obj.obj (op (⋯.pullback (f j)))) → yoneda.obj (⋯.pullback (f j)) ⟶ F.obj)
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (F.obj (P := Presheaf.IsSheaf zariskiTopology).map (⋯.fst' (f j)).op) :
              (a : F.obj.obj (op (X i))) → F.obj.obj (op (⋯.pullback (f j))))
            ((yonedaEquiv : (a : yoneda.obj (X i) ⟶ F.obj) → F.obj.obj (op (X i))) (f i) :
              F.obj (P := Presheaf.IsSheaf zariskiTopology).obj (op (X i))) :
          F.obj (P := Presheaf.IsSheaf zariskiTopology).obj (op (⋯.pullback (f j)))) :
      Functor.obj yoneda (⋯.pullback (f j)) ⟶ F.obj (P := Presheaf.IsSheaf zariskiTopology))
    ((Equiv.symm yonedaEquiv : (a : F.obj.obj (op (⋯.pullback (f j)))) → yoneda.obj (⋯.pullback (f j)) ⟶ F.obj)
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (F.obj (P := Presheaf.IsSheaf zariskiTopology).map ((⋯.fst' (f i)).op ≫ (⋯.symmetry ⋯).op)) :
              (a : F.obj.obj (op (X j))) → F.obj.obj (op (⋯.pullback (f j))))
            ((yonedaEquiv : (a : yoneda.obj (X j) ⟶ F.obj) → F.obj.obj (op (X j))) (f j) :
              F.obj (P := Presheaf.IsSheaf zariskiTopology).obj (op (X j))) :
          F.obj (P := Presheaf.IsSheaf zariskiTopology).obj (op (⋯.pullback (f j)))) :
      Functor.obj yoneda (⋯.pullback (f j)) ⟶ F.obj (P := Presheaf.IsSheaf zariskiTopology))
have a₂ : CategoryTheory.Functor.map yoneda (sorry.fst' (f j)) ≫ f i =
  (Equiv.symm yonedaEquiv : (a : F.obj.obj (op (sorry.pullback (f j)))) → yoneda.obj (sorry.pullback (f j)) ⟶ F.obj)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.obj (P := Presheaf.IsSheaf zariskiTopology).map (sorry.symmetry sorry).op) :
          (a : F.obj.obj (op (sorry.pullback (f i)))) → F.obj.obj (op (sorry.pullback (f j))))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (F.obj (P := Presheaf.IsSheaf zariskiTopology).map (sorry.fst' (f i)).op) :
              (a : F.obj.obj (op (X j))) → F.obj.obj (op (sorry.pullback (f i))))
            ((yonedaEquiv : (a : yoneda.obj (X j) ⟶ F.obj) → F.obj.obj (op (X j))) (f j) :
              F.obj (P := Presheaf.IsSheaf zariskiTopology).obj (op (X j))) :
          F.obj (P := Presheaf.IsSheaf zariskiTopology).obj (op (sorry.pullback (f i)))) :
      F.obj (P := Presheaf.IsSheaf zariskiTopology).obj (op (sorry.pullback (f j))))
Tactic: rw [yonedaEquiv_naturality, Equiv.symm_apply_apply,
        Functor.map_comp_apply, yonedaEquiv_naturality, yonedaEquiv_naturality,
        Equiv.symm_apply_apply, ← Functor.map_comp_assoc,
        Functor.relativelyRepresentable.symmetry_fst, ((hf i).rep.isPullback' (f j)).w]