on the pointfree counterpart of the local definition of classical continuous maps

The familiar classical result that a continuous map from a space x to a space y can be defined by giving continuous maps φu:u→y on each member u of an open cover c of x such that φu∣u∩v=φv∣u∩v for all u,v∈c was recently shown to have an exact analogue in pointfree topology, and the same was done for the familiar classical counterpart concerning finite closed covers of a space x (picado and pultr [4]). this note presents alternative proofs of these pointfree results which differ from those of [4] by treating the issue in terms of frame homomorphisms while the latter deals with the dual situation concerning localic maps. a notable advantage of the present approach is that it also provides proofs of the analogous results for some significant variants of frames which are not covered by the localic arguments.