  The goal of the present paper is to establish some kind of regularity of an
energy minimizer map between Riemannian polyhedra. More precisely, we will show
the h\"{o}lder continuity of local energy minimizers between Riemannian
polyhedra with the target spaces without focal points. With this new result, we
also complete our existence theorem obtained in [5], and consequently we
generalize completely, to the case of target polyhedra without focal points
(which is weaker geometric condition than the nonpositivity of the curvature),
the Eells-fuglede's existence and regularity theorem [12, chapters 10, 11]
which is the new version of the famous Eells-Sampson's theorem [13].
