Let X and X be compact Hausdorff spaces, and X , X be Banach lattices. Let X denote the Banach lattice of all continuous X -valued functions on X equipped with the pointwise ordering and the sup norm. We prove that if there exists a Riesz isomorphism X such that X is non-vanishing on X if and only if X is non-vanishing on X , then X is homeomorphic to X , and X is Riesz isomorphic to X . In this case, X can be written as a weighted composition operator: X , where X is a homeomorphism from X onto X , and X is a Riesz isomorphism from X onto X for every X in X . This generalizes some known results obtained recently.