def METHOD_NAME(self, rename_vertices=True):
    r"""
    Return the diagonal morphism in `Hom(S, S \times S)`.
    EXAMPLES::
        sage: S = simplicial_complexes.Sphere(2)
        sage: H = Hom(S,S.product(S, is_mutable=False))
        sage: d = H.diagonal_morphism(); d
        Simplicial complex morphism:
          From: Minimal triangulation of the 2-sphere
          To:   Simplicial complex with 16 vertices and 96 facets
          Defn: 0 |--> L0R0
                1 |--> L1R1
                2 |--> L2R2
                3 |--> L3R3
        sage: T = SimplicialComplex([[0], [1]], is_mutable=False)
        sage: U = T.product(T, rename_vertices=False, is_mutable=False)
        sage: G = Hom(T, U)
        sage: e = G.diagonal_morphism(rename_vertices=False); e
        Simplicial complex morphism:
          From: Simplicial complex with vertex set (0, 1) and facets {(0,), (1,)}
          To:   Simplicial complex with 4 vertices and
                facets {((0, 0),), ((0, 1),), ((1, 0),), ((1, 1),)}
          Defn: 0 |--> (0, 0)
                1 |--> (1, 1)
    """
    # Preserve whether the codomain is mutable when renaming the vertices.
    mutable = self._codomain.is_mutable()
    X = self._domain.product(self._domain, rename_vertices=rename_vertices,
                             is_mutable=mutable)
    if self._codomain != X:
        raise TypeError("diagonal morphism is only defined for Hom(X,XxX)")
    f = {}
    if rename_vertices:
        f = {i: "L{0}R{0}".format(i) for i in self._domain.vertices()}
    else:
        f = {i: (i, i) for i in self._domain.vertices()}
    return SimplicialComplexMorphism(f, self._domain, X)