∀ {R S : CommRingCat} (f : R ⟶ S) (algebraizeInst : Module.FaithfullyFlat ↑R ↑S),
  CategoryTheory.CategoryStruct.comp
      (RingEquiv.ofBijective (Algebra.codRestrictEqLocusPushoutCocone ↑R ↑S) ⋯).toCommRingCatIso.hom
      (CommRingCat.equalizerFork (CommRingCat.pushoutCocone ↑R ↑S ↑S).inl (CommRingCat.pushoutCocone ↑R ↑S ↑S).inr).ι =
    CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Fork.ofι (CommRingCat.ofHom (algebraMap ↑R ↑S)) ⋯).ι
      (CategoryTheory.Iso.refl (CommRingCat.of ↑S)).hom