Excisive triads and double mapping cylinders

Given a topological space X together with subspaces X0 and X1 such that X=X0∪X1 and setting X2=X0∩X1, there is a quotient map π:M→X, where M is the double mapping cylinder of the cotriad X0←X2→X1. We prove that if (X;X0,X1) is excisive, then π is a weak equivalence. We use this fact to give a correct proof of a theorem on maps of excisive triads due to J.P. May.