Lefschetz number of a doubled map ($L(F)=2L(f)-L(f{|}_{\partial N})$)

Let $F$ be the self-map of the double $D(N)=N{\cup }_{\partial N}N$ induced by a self-map $f$ preserving the boundary. Naturality of the Mayer–Vietoris sequence and alternating-trace additivity give