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.