∀ (x : LightProfinite),
  CategoryTheory.CategoryStruct.comp
      (lightProfiniteToLightDiagram.map
        ((CategoryTheory.Iso.refl (CategoryTheory.Functor.id LightProfinite)).hom.app x))
      ((CategoryTheory.NatIso.ofComponents
              (fun x =>
                lightDiagramToProfinite.preimageIso
                  (CategoryTheory.Iso.refl
                    (lightDiagramToProfinite.obj
                      ((lightDiagramToLightProfinite.comp lightProfiniteToLightDiagram).obj x))))
              @LightProfinite.equivDiagram._proof_1).hom.app
        (lightProfiniteToLightDiagram.obj x)) =
    CategoryTheory.CategoryStruct.id (lightProfiniteToLightDiagram.obj x)