A supplement to “On the Fourier coefficients of Hilbert modular forms of half-integral weight over arbitrary algebraic number fields, Tsukuba J. Math. 37(2013), 1-11”

The purpose of this note is to prove that the Shintani lift of Hilbert modular forms over algebraic number fields commutes with the action of Hecke operators. We show our assertion using the commutativity of the Shimura lift with Hecke operators and some properties of adjoint mappings. This commutativity of the Shintani lift plays an essential role for the proof of the Waldspurger-type theorem concerning the Fourier coefficients of modular forms of half-integral weight over arbitrary algebraic number fields.