\textbf{Investigation of the magnetic properties in double perovskite R}$_{\mathrm{\mathbf{2}}}$\textbf{CoMnO}$_{\mathrm{\mathbf{6}}}$\textbf{ single crystals (R}$=$\textbf{rare earth: La to Lu)}

We have successfully synthesized the series of the double-perovskite R$_{\mathrm{2}}$CoMnO$_{\mathrm{6}}$ (R$=$rare earth: La to Lu) single crystals and have investigated their magnetic properties. The ferromagnetic order of Co$^{\mathrm{2+}}$/Mn$^{\mathrm{4+}}$ 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 La$_{\mathrm{2}}$CoMnO$_{\mathrm{6}}$ to 48 K for Lu$_{\mathrm{2}}$CoMnO$_{\mathrm{6}}$, 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.