Let S₁={∞} and S₂={w: P_s(w)=0}, P_s(w) being a uniqueness polynomial under some restricted conditions. Then, for any given nonconstant meromorphic function f, there exist at most finitely many nonconstant meromorphic functions g such that f^-1(S_i)=g^-1(S_i), (i=1,2), where f^-1(S_i) and g^-1(S_i)$ denote the pull-backs of S_i considered as a divisor, namely, the inverse images of S_i counted with multiplicities, by f and g respectively.