We have successfully synthesized the series of the double-perovskite R2CoMnO6 (R = rare earth: La to Lu) single crystals and have investigated their magnetic properties. The ferromagnetic order of Co(2+)/Mn(4+) spins emerges mainly along the c axis. Upon decreasing the size of rare earth ion, the magnetic transition temperature decreases linearly from 204 K for La2CoMnO6 to 48 K for Lu2CoMnO6, along with the enhancement of monoclinic distortion. The temperature and magnetic-field dependences of magnetization reveal the various magnetic characteristics such as the metamagnetic transition in R = Eu, the isotropic nature of rare earth moment in R = Gd, and the reversal of magnetic anisotropy in R = Tb and Dy. Our results offer comprehensive information for understanding the roles of mixed-valent magnetic ions and rare earth magnetic moments on the magnetic properties.