As shown by Dinh The Luc in his well-known monograph on vector optimization, the cone-quasiconvex vector-valued functions can be characterized in terms of scalar quasiconvexity by means of the smallest strictly monotonic functions. The aim of this paper is to show that similar characterizations can be given for the general class of $(\Gamma,\Omega)$-quasiconvex functions, this time by means of the so-called properly characteristic functions associated to the binary relation $\Omega$.